TangentFrom Wikipedia, the free encyclopedia

Contents

1 Curve 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Conventions and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Lengths of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Algebraic curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Derivative 102.1 Differentiation and derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Rigorous definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Definition over the hyperreals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.5 Continuity and differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.6 The derivative as a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.7 Higher derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.8 Inflection point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Notation (details) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Leibniz’s notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Lagrange’s notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 Newton’s notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Euler’s notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Rules of computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Rules for basic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Rules for combined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Computation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Derivatives in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 Derivatives of vector valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

i

ii CONTENTS

2.4.2 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.3 Directional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.4 Total derivative, total differential and Jacobian matrix . . . . . . . . . . . . . . . . . . . . 24

2.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.9.1 Print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.9.2 Online books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.9.3 Web pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Tangent 303.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Tangent line to a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Analytical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.3 Normal line to a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.4 Angle between curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.5 Multiple tangents at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Tangent circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Surfaces and higher-dimensional manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 1

Curve

For other uses, see Curve (disambiguation).In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line

but which is not required to be straight. This entails that a line is a special case of curve, namely a curve with nullcurvature.[lower-alpha 1] Often curves in two-dimensional (plane curves) or three-dimensional (space curves) Euclideanspace are of interest.Various disciplines within mathematics have given the term different meanings depending on the area of study, sothe precise meaning depends on context. However, many of these meanings are special instances of the definitionwhich follows. A curve is a topological space which is locally homeomorphic to a line. In everyday language, thismeans that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simpleexample of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiplemathematical fields.Closely related meanings are "graph of a function" (as in "Phillips curve") and "two-dimensional or three-dimensionalgraph without a kink”.In non-mathematical language, the term is often used metaphorically, as in "learning curve".

1.1 History

Interest in curves began long before they were the subject of mathematical study. This can be seen in numerousexamples of their decorative use in art and on everyday objects dating back to prehistoric times.[1] Curves, or at leasttheir graphical representations, are simple to create, for example by a stick in the sand on a beach.Historically, the term “line” was used in place of the more modern term “curve”. Hence the phrases “straight line”and “right line” were used to distinguish what are today called lines from “curved lines”. For example, in Book Iof Euclid’s Elements, a line is defined as a “breadthless length” (Def. 2), while a straight line is defined as “a linethat lies evenly with the points on itself” (Def. 4). Euclid’s idea of a line is perhaps clarified by the statement “Theextremities of a line are points,” (Def. 3).[2] Later commentators further classified lines according to various schemes.For example:[3]

• Composite lines (lines forming an angle)

• Incomposite lines

• Determinate (lines that do not extend indefinitely, such as the circle)• Indeterminate (lines that extend indefinitely, such as the straight line and the parabola)

The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometricalproblems that could not be solved using standard compass and straightedge construction. These curves include:

• The conic sections, deeply studied by Apollonius of Perga

• The cissoid of Diocles, studied by Diocles and used as a method to double the cube.[4]

1

2 CHAPTER 1. CURVE

A parabola, a simple example of a curve

• The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect anangle.[5]

• The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle.[6]

• The spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius.

A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century. Thisenabled a curve to be described using an equation rather than an elaborate geometrical construction. This not onlyallowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves thatcan be defined using algebraic equations, algebraic curves, and those that cannot, transcendental curves. Previously,

1.1. HISTORY 3

Megalithic art from Newgrange showing an early interest in curves

The curves created by slicing a cone (conic sections) were among the curves studied in ancient Greece.

curves had been described as “geometrical” or “mechanical” according to how they were, or supposedly could be,generated.[1]

Conic sections were applied in astronomy by Kepler. Newton also worked on an early example in the calculus of

4 CHAPTER 1. CURVE

1

0

−1

−2

x

2

−2 −1 0 1 2

y

Analytic geometry allowed curves, such as the Folium of Descartes, to be defined using equations instead of geometrical construction.

variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introducedproperties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problemof a hanging chain, the sort of question that became routinely accessible by means of differential calculus.In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studiedthe cubic curves, in the general description of the real points into 'ovals’. The statement of Bézout’s theorem showeda number of aspects which were not directly accessible to the geometry of the time, to do with singular points andcomplex solutions.From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example theJordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of thespace-filling curves finally provoked the modern definitions of curve.

1.2 Topology

In topology, a curve is defined as follows. Let I be an interval of real numbers (i.e. a non-empty connected subsetof R ). Then a curve γ is a continuous mapping γ : I → X , where X is a topological space.

• The curve γ is said to be simple, or a Jordan arc, if it is injective, i.e. if for all x , y in I , we have γ(x) = γ(y)

1.2. TOPOLOGY 5

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1.5 -1 -0.5 0 0.5 1

c.im

c.re

Boundaries of 53 hyperbolic components of Mandelbrot set made in 13sec

one period 1 component = {c:c=(2*w-w*w)/4} one period 2 component = {c:c=(w/4 -1)}

three period 3 components (blue)six period 4 components (magenta)fifteen period 5 components (black)

27 period 6 components (black)

Boundaries of hyperbolic components of Mandelbrot set as closed curves

implies x = y . If I is a closed bounded interval [a, b] , we also allow the possibility γ(a) = γ(b) (thisconvention makes it possible to talk about “closed” simple curves, see below).

In other words, this curve “does not cross itself and has no missing points”.[7]

• If γ(x) = γ(y) for some x = y (other than the extremities of I ), then γ(x) is called a double (or multiple)point of the curve.

• A curve γ is said to be closed or a loop if I = [a, b] and if γ(a) = γ(b) . A closed curve is thus a continuousmapping of the circle S1 ; a simple closed curve is also called a Jordan curve. The Jordan curve theoremstates that such curves divide the plane into an “interior” and an “exterior”.

A plane curve is a curve for which X is the Euclidean plane—these are the examples first encountered—or in somecases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space;a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below).However, in the case of algebraic curves it is very common to consider number systems more general than the reals.This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure thatis “like” a line, without thickness and drawn without interruption, although it also includes figures that can hardly

6 CHAPTER 1. CURVE

be called curves in common usage. For example, the image of a curve can cover a square in the plane (space-fillingcurve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and evenpositive Lebesgue measure[8] (the last example can be obtained by small variation of the Peano curve construction).The dragon curve is another unusual example.

1.3 Conventions and terminology

The distinction between a curve and its image is important. Two distinct curves may have the same image. Forexample, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times.Many times, however, we are just interested in the image of the curve. It is important to pay attention to context andconvention in reading.Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and “curve”for what we are calling the image of a curve. The term “curve” is more common in vector calculus and differentialgeometry.An arc or segment of a curve is a part of a curve that is bounded by two distinct end points and contains every pointon the curve between its end points. Depending on how the arc is defined, either of the two end points may or maynot be part of it. When the arc is straight, it is typically called a line segment (see also chord).

1.4 Lengths of curves

Main article: Arc length

If X is a metric space with metric d , then we can define the length of a curve γ : [a, b] → X by

length(γ) = sup{

n∑i=1

d(γ(ti), γ(ti−1)) : n ∈ N and a = t0 < t1 < · · · < tn = b

}

where the sup is over all n and all partitions t0 < t1 < · · · < tn of [a, b] .A rectifiable curve is a curve with finite length. A parametrization of γ is called natural (or unit speed orparametrised by arc length) if for any t1, t2 ∈ [a, b] , we have

length(γ|[t1,t2]) = |t2 − t1|

If γ is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define thespeed (or metric derivative) of γ at t0 as

speed(t0) = lim supt→t0

d(γ(t), γ(t0))

|t− t0|

and then

length(γ) =∫ b

a

speed(t) dt

In particular, if X = Rn is an Euclidean space and γ : [a, b] → Rn is differentiable then

length(γ) =∫ b

a

|γ′(t)| dt

1.5. DIFFERENTIAL GEOMETRY 7

1.5 Differential geometry

Main article: Differential geometry of curves

While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines intwo-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. Theneeds of geometry, and also for example classical mechanics are to have a notion of curve in space of any number ofdimensions. In general relativity, a world line is a curve in spacetime.If X is a differentiable manifold, then we can define the notion of differentiable curve in X . This general idea isenough to cover many of the applications of curves in mathematics. From a local point of view one can take X to beEuclidean space. On the other hand, it is useful to be more general, in that (for example) it is possible to define thetangent vectors to X by means of this notion of curve.If X is a smooth manifold, a smooth curve in X is a smooth map

γ : I → X

This is a basic notion. There are less and more restricted ideas, too. If X is a Ck manifold (i.e., a manifold whosecharts are k times continuously differentiable), then a Ck curve in X is such a curve which is only assumed to beCk (i.e. k times continuously differentiable). If X is an analytic manifold (i.e. infinitely differentiable and charts areexpressible as power series), and γ is an analytic map, then γ is said to be an analytic curve.A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows toa stop or backtracks on itself.) Two Ck differentiable curves

γ1 : I → X

γ2 : J → X

are said to be equivalent if there is a bijective Ck map

p : J → I

such that the inverse map

p−1 : I → J

is also Ck , and

γ2(t) = γ1(p(t))

for all t . The map γ2 is called a reparametrisation of γ1 ; and this makes an equivalence relation on the set of all Ck

differentiable curves in X . A Ck arc is an equivalence class of Ck curves under the relation of reparametrisation.

1.6 Algebraic curve

Main article: Algebraic curve

Algebraic curves are the curves considered in algebraic geometry. A plane algebraic curve is the locus of the pointsof coordinates x, y such that f(x, y) = 0, where f is a polynomial in two variables defined over some field F. Alge-braic geometry normally looks not only on points with coordinates in F but on all the points with coordinates in analgebraically closed field K. If C is a curve defined by a polynomial f with coefficients in F, the curve is said defined

8 CHAPTER 1. CURVE

over F. The points of the curve C with coordinates in a field G are said rational over G and can be denoted C(G));thus the full curve C = C(K).Algebraic curves can also be space curves, or curves in even higher dimension, obtained as the intersection (commonsolution set) of more than one polynomial equation in more than two variables. By eliminating variables (by anytool of elimination theory), an algebraic curve may be projected onto a plane algebraic curve, which however mayintroduce singularities such as cusps or double points.A plane curve may also be completed in a curve in the projective plane: if a curve is defined by a polynomial f oftotal degree d, then wdf(u/w, v/w) simplifies to a homogeneous polynomial g(u, v, w) of degree d. The values of u,v, w such that g(u, v, w) = 0 are the homogeneous coordinates of the points of the completion of the curve in theprojective plane and the points of the initial curve are those such w is not zero. An example is the Fermat curve un

+ vn = wn, which has an affine form xn + yn = 1. A similar process of homogenization may be defined for curves inhigher dimensional spacesImportant examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero,and elliptic curves, which are nonsingular curves of genus one studied in number theory and which have importantapplications to cryptography. Because algebraic curves in fields of characteristic zero are most often studied overthe complex numbers, algebraic curves in algebraic geometry may be considered as real surfaces. In particular, thenon-singular complex projective algebraic curves are called Riemann surfaces.

1.7 See also

• Curvature

• Curve orientation

• Curve sketching

• Differential geometry of curves

• Gallery of curves

• implicit curve

• List of curves topics

• List of curves

• Osculating circle

• Parametric surface

• Path (topology)

• Position vector

• Vector-valued function

• Curve fitting

1.8 Notes[1] In current language, a line is typically required to be straight. Historically, however, lines could be “curved” or “straight”.

1.9 References[1] Lockwood p. ix

[2] Heath p. 153

1.10. EXTERNAL LINKS 9

[3] Heath p. 160

[4] Lockwood p. 132

[5] Lockwood p. 129

[6] O'Connor, John J.; Robertson, Edmund F., “Spiral of Archimedes”, MacTutor History of Mathematics archive, Universityof St Andrews.

[7] “Jordan arc definition at Dictionary.com. Dictionary.com Unabridged. Random House, Inc”. Dictionary.reference.com.Retrieved 2012-03-14.

[8] Osgood, William F. (January 1903). “A Jordan Curve of Positive Area”. Transactions of the American MathematicalSociety (American Mathematical Society) 4 (1): 107–112. doi:10.2307/1986455. ISSN 0002-9947. JSTOR 1986455.

• A.S. Parkhomenko (2001), “Line (curve)", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• B.I. Golubov (2001), “Rectifiable curve”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Euclid, commentary and trans. by T. L. Heath Elements Vol. 1 (1908 Cambridge) Google Books

• E. H. Lockwood A Book of Curves (1961 Cambridge)

1.10 External links• Famous Curves Index, School of Mathematics and Statistics, University of St Andrews, Scotland

• Mathematical curves A collection of 874 two-dimensional mathematical curves

• Gallery of Space Curves Made from Circles, includes animations by Peter Moses

• Gallery of Bishop Curves and Other Spherical Curves, includes animations by Peter Moses

• The Encyclopedia of Mathematics article on lines.

• The Manifold Atlas page on 1-manifolds.

Chapter 2

Derivative

This article is about the term as used in calculus. For a less technical overview of the subject, see differential calculus.For other uses, see Derivative (disambiguation).

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal tothe derivative of the function at the marked point.

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value ordependent variable) which is determined by another quantity (the independent variable). Derivatives are a funda-mental tool of calculus. For example, the derivative of the position of a moving object with respect to time is theobject’s velocity: this measures how quickly the position of the object changes when time is advanced.The derivative of a function of a single variable at a chosen input value is the slope of the tangent line to the graphof the function at that point. This means that it describes the best linear approximation of the function near thatinput value. For this reason, the derivative is often described as the “instantaneous rate of change”, the ratio of theinstantaneous change in the dependent variable to that of the independent variable.Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinter-

10

2.1. DIFFERENTIATION AND DERIVATIVE 11

preted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation tothe graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation withrespect to the basis given by the choice of independent and dependent variables. It can be calculated in terms ofthe partial derivatives with respect to the independent variables. For a real-valued function of several variables, theJacobian matrix reduces to the gradient vector.The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. Thefundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and inte-gration constitute the two fundamental operations in single-variable calculus.[1]

2.1 Differentiation and derivative

Differentiation is the action of computing a derivative. The derivative of a function f(x) of a variable x is a measure ofthe rate at which the value of the function changes with respect to the change of the variable. It is called the derivativeof f with respect to x. If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slopeof this graph at each point.The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning thatthe graph of y divided by x is a line. In this case, y = f(x) = m x + b, for real numbers m and b, and the slope m isgiven by

m =in changeyin changex =

∆y

∆x,

where the symbol Δ (Delta) is an abbreviation for “change in.” This formula is true because

y +∆y = f (x+∆x) = m (x+∆x) + b = mx+m∆x+ b = y +m∆x.

Thus, since

y +∆y = y +m∆x,

it follows that

∆y = m∆x.

This gives an exact value for the slope of a line. If the function f is not linear (i.e. its graph is not a straight line),however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value forthis rate of change at any given value of x.Rate of change as a limit value

x

tangent line

slope= f'(x)

Figure 1. The tangent line at (x, f(x))

12 CHAPTER 2. DERIVATIVE

x

secant line

f(x)

f(x+h)

x+h

Figure 2. The secant to curve y= f(x) determined by points (x, f(x)) and (x+h, f(x+h))

x+hx+h'x+h"x

Figure 3. The tangent line as limit of secants

Figure 4. Animated illustration: the tangent line (derivative) as the limit of secants

The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differencesΔy / Δx as Δx becomes infinitely small.

2.1.1 Notation

Two distinct notations are commonly used for the derivative, one deriving from Leibniz and the other from JosephLouis Lagrange.In Leibniz’s notation, an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written

dy

dx

suggesting the ratio of two infinitesimal quantities. (The above expression is read as “the derivative of y with respectto x", “d y by d x”, or “d y over d x”. The oral form “d y d x” is often used conversationally, although it may lead toconfusion.)In Lagrange’s notation, the derivative with respect to x of a function f(x) is denoted f ' (x) (read as “f prime of x”) orfx ' (x) (read as “f prime x of x”), in case of ambiguity of the variable implied by the derivation. Lagrange’s notationis sometimes incorrectly attributed to Newton.

2.1. DIFFERENTIATION AND DERIVATIVE 13

2.1.2 Rigorous definition

The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit ofdifference quotients of real numbers.[2] This is the approach described below.Let f be a real valued function defined in an open neighborhood of a real number a. In classical geometry, the tangentline to the graph of the function f at a was the unique line through the point (a, f(a)) that did not meet the graph off transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x ata is, geometrically, the slope of the tangent line to the graph of f at (a, f(a)). The slope of the tangent line is veryclose to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). Theselines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line,and smaller values (in absolute value) of h will, in general, give better approximations. The slope m of the secant lineis the difference between the y values of these points divided by the difference between the x values, that is,

m =∆f(a)

∆a=

f(a+ h)− f(a)

(a+ h)− (a)=

f(a+ h)− f(a)

h.

This expression is Newton's difference quotient. Passing from an approximation to an exact answer is done using alimit. Geometrically, the limit of the secant lines is the tangent line. Therefore, the limit of the difference quotientas h approaches zero, if it exists, should represent the slope of the tangent line to (a, f(a)). This limit is defined to bethe derivative of the function f at a:

A secant approaches a tangent when ∆x → 0 .

f ′(a) = limh→0

f(a+ h)− f(a)

h.

When the limit exists, f is said to be differentiable at a. Here f′ (a) is one of several common notations for thederivative (see below).

14 CHAPTER 2. DERIVATIVE

Equivalently, the derivative satisfies the property that

limh→0

f(a+ h)− f(a)− f ′(a) · hh

= 0,

which has the intuitive interpretation (see Figure 1) that the tangent line to f at a gives the best linear approximation

f(a+ h) ≈ f(a) + f ′(a)h

to f near a (i.e., for small h). This interpretation is the easiest to generalize to other settings (see below).Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be founddirectly using this method. Instead, define Q(h) to be the difference quotient as a function of h:

Q(h) =f(a+ h)− f(a)

h.

Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). If f is a continuous function, meaning thatits graph is an unbroken curve with no gaps, then Q is a continuous function away from h = 0. If the limit limh→₀Q(h)exists, meaning that there is a way of choosing a value for Q(0) that makes Q a continuous function, then the functionf is differentiable at a, and its derivative at a equals Q(0).In practice, the existence of a continuous extension of the difference quotient Q(h) to h = 0 is shown by modifyingthe numerator to cancel h in the denominator. Such manipulations can make the limit value of Q for small h cleareven though Q is still not defined at h = 0. This process can be long and tedious for complicated functions, and manyshortcuts are commonly used to simplify the process.

2.1.3 Definition over the hyperreals

Relative to a hyperreal extension R ⊂ R* of the real numbers, the derivative of a real function y = f(x) at a real point xcan be defined as the shadow of the quotient ∆y/∆x for infinitesimal ∆x, where ∆y = f(x+ ∆x) - f(x). Here the naturalextension of f to the hyperreals is still denoted f. Here the derivative is said to exist if the shadow is independent ofthe infinitesimal chosen.

2.1.4 Example

The squaring function f(x) = x2 is differentiable at x = 3, and its derivative there is 6. This result is established bycalculating the limit as h approaches zero of the difference quotient of f(3):

f ′(3) = limh→0

f(3 + h)− f(3)

h= lim

h→0

(3 + h)2 − 32

h

= limh→0

9 + 6h+ h2 − 9

h= lim

h→0

6h+ h2

h= lim

h→0(6 + h).

The last expression shows that the difference quotient equals 6 + h when h ≠ 0 and is undefined when h = 0, becauseof the definition of the difference quotient. However, the definition of the limit says the difference quotient does notneed to be defined when h = 0. The limit is the result of letting h go to zero, meaning it is the value that 6 + h tendsto as h becomes very small:

limh→0

(6 + h) = 6 + 0 = 6.

Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so its derivative at x = 3 is f′(3) = 6.More generally, a similar computation shows that the derivative of the squaring function at x = a is f′(a) = 2a.

2.1. DIFFERENTIATION AND DERIVATIVE 15

This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jumpdiscontinuity).

2.1.5 Continuity and differentiability

If y = f(x) is differentiable at a, then f must also be continuous at a. As an example, choose a point a and let f bethe step function that returns a value, say 1, for all x less than a, and returns a different value, say 10, for all x greaterthan or equal to a. f cannot have a derivative at a. If h is negative, then a + h is on the low part of the step, so thesecant line from a to a + h is very steep, and as h tends to zero the slope tends to infinity. If h is positive, then a + his on the high part of the step, so the secant line from a to a + h has slope zero. Consequently, the secant lines do notapproach any single slope, so the limit of the difference quotient does not exist.[3]

However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolutevalue function y = | x | is continuous at x = 0, but it is not differentiable there. If h is positive, then the slope of thesecant line from 0 to h is one, whereas if h is negative, then the slope of the secant line from 0 to h is negative one.This can be seen graphically as a “kink” or a “cusp” in the graph at x = 0. Even a function with a smooth graph is notdifferentiable at a point where its tangent is vertical: For instance, the function y = x1/3 is not differentiable at x = 0.In summary: for a function f to have a derivative it is necessary for the function f to be continuous, but continuityalone is not sufficient.Most functions that occur in practice have derivatives at all points or at almost every point. Early in the history ofcalculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mildconditions, for example if the function is a monotone function or a Lipschitz function, this is true. However, in 1872Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. Thisexample is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that havea derivative at some point is a meager set in the space of all continuous functions.[4] Informally, this means that hardlyany continuous functions have a derivative at even one point.

16 CHAPTER 2. DERIVATIVE

1

2

3

4

−3 −2 −1 1 2 30

y = |x|

The absolute value function is continuous, but fails to be differentiable at x = 0 since the tangent slopes do not approach the samevalue from the left as they do from the right.

2.1.6 The derivative as a function

Let f be a function that has a derivative at every point a in the domain of f. Because every point a has a derivative,there is a function that sends the point a to the derivative of f at a. This function is written f ′(x) and is called thederivative function or the derivative of f. The derivative of f collects all the derivatives of f at all the points in thedomain of f.Sometimes f has a derivative at most, but not all, points of its domain. The function whose value at a equals f ′(a)whenever f ′(a) is defined and elsewhere is undefined is also called the derivative of f. It is still a function, but itsdomain is strictly smaller than the domain of f.Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is theset of all functions that have derivatives at every point of their domain and whose range is a set of functions. If wedenote this operator by D, then D(f) is the function f ′(x). Since D(f) is a function, it can be evaluated at a point a.By the definition of the derivative function, D(f)(a) = f ′(a).For comparison, consider the doubling function f(x) = 2x; f is a real-valued function of a real number, meaning thatit takes numbers as inputs and has numbers as outputs:

1 7→ 2,

2 7→ 4,

3 7→ 6.

The operator D, however, is not defined on individual numbers. It is only defined on functions:

D(x 7→ 1) = (x 7→ 0),

D(x 7→ x) = (x 7→ 1),

D(x 7→ x2) = (x 7→ 2 · x).

Because the output of D is a function, the output of D can be evaluated at a point. For instance, when D is applied to

2.1. DIFFERENTIATION AND DERIVATIVE 17

the squaring function, x ↦ x2, D outputs the doubling function x ↦ 2x, which we named f(x). This output functioncan then be evaluated to get f(1) = 2, f(2) = 4, and so on.

2.1.7 Higher derivatives

Let f be a differentiable function, and let f ′(x) be its derivative. The derivative of f ′(x) (if it has one) is written f′′(x) and is called the second derivative of f. Similarly, the derivative of a second derivative, if it exists, is writtenf ′′′(x) and is called the third derivative of f. Continuing this process, one can define, if it exists, the nth derivativeas the derivative of the (n−1)th derivative. These repeated derivatives are called higher-order derivatives. The nthderivative is also called the derivative of order n.If x(t) represents the position of an object at time t, then the higher-order derivatives of x have physical interpretations.The second derivative of x is the derivative of x′(t), the velocity, and by definition this is the object’s acceleration.The third derivative of x is defined to be the jerk, and the fourth derivative is defined to be the jounce.A function f need not have a derivative, for example, if it is not continuous. Similarly, even if f does have a derivative,it may not have a second derivative. For example, let

f(x) =

{+x2, ifx ≥ 0

−x2, ifx ≤ 0.

Calculation shows that f is a differentiable function whose derivative is

f ′(x) =

{+2x, ifx ≥ 0

−2x, ifx ≤ 0.

f ′(x) is twice the absolute value function, and it does not have a derivative at zero. Similar examples show that afunction can have k derivatives for any non-negative integer k but no (k + 1)th-order derivative. A function that has ksuccessive derivatives is called k times differentiable. If in addition the kth derivative is continuous, then the functionis said to be of differentiability class Ck. (This is a stronger condition than having k derivatives. For an example, seedifferentiability class.) A function that has infinitely many derivatives is called infinitely differentiable or smooth.On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polyno-mial of degree n is differentiated n times, then it becomes a constant function. All of its subsequent derivatives areidentically zero. In particular, they exist, so polynomials are smooth functions.The derivatives of a function f at a point x provide polynomial approximations to that function near x. For example,if f is twice differentiable, then

f(x+ h) ≈ f(x) + f ′(x)h+ 12f

′′(x)h2

in the sense that

limh→0

f(x+ h)− f(x)− f ′(x)h− 12f

′′(x)h2

h2= 0.

If f is infinitely differentiable, then this is the beginning of the Taylor series for f evaluated at x + h around x.

2.1.8 Inflection point

Main article: Inflection point

A point where the second derivative of a function changes sign is called an inflection point.[5] At an inflection point,the second derivative may be zero, as in the case of the inflection point x = 0 of the function y = x3, or it may fail toexist, as in the case of the inflection point x = 0 of the function y = x1/3. At an inflection point, a function switchesfrom being a convex function to being a concave function or vice versa.

18 CHAPTER 2. DERIVATIVE

2.2 Notation (details)

Main article: Notation for differentiation

2.2.1 Leibniz’s notation

Main article: Leibniz’s notation

The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. It is still commonly used when theequation y = f(x) is viewed as a functional relationship between dependent and independent variables. Then the firstderivative is denoted by

dy

dx,

df

dx(x), or d

dxf(x),

and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using the notation

dny

dxn,

dnf

dxn(x), or dn

dxnf(x)

for the nth derivative of y = f(x) (with respect to x). These are abbreviations for multiple applications of the derivativeoperator. For example,

d2y

dx2=

d

dx

(dy

dx

).

With Leibniz’s notation, we can write the derivative of y at the point x = a in two different ways:

dy

dx

∣∣∣∣x=a

=dy

dx(a).

Leibniz’s notation allows one to specify the variable for differentiation (in the denominator). This is especially relevantfor partial differentiation. It also makes the chain rule easy to remember:[6]

dy

dx=

dy

du· dudx

.

2.2.2 Lagrange’s notation

Sometimes referred to as prime notation,[7] one of the most common modern notation for differentiation is due toJoseph-Louis Lagrange and uses the prime mark, so that the derivative of a function f(x) is denoted f′(x) or simplyf′. Similarly, the second and third derivatives are denoted

(f ′)′ = f ′′ and (f ′′)′ = f ′′′.

To denote the number of derivatives beyond this point, some authors use Roman numerals in superscript, whereasothers place the number in parentheses:

f iv or f (4).

The latter notation generalizes to yield the notation f (n) for the nth derivative of f – this notation is most usefulwhen we wish to talk about the derivative as being a function itself, as in this case the Leibniz notation can becomecumbersome.

2.2. NOTATION (DETAILS) 19

2.2.3 Newton’s notation

Newton’s notation for differentiation, also called the dot notation, places a dot over the function name to represent atime derivative. If y = f(t), then

y and y

denote, respectively, the first and second derivatives of y with respect to t. This notation is used exclusively for timederivatives, meaning that the independent variable of the function represents time. It is very common in physicsand in mathematical disciplines connected with physics such as differential equations. While the notation becomesunmanageable for high-order derivatives, in practice only very few derivatives are needed.

Fluent and fluxions

Newton tried to explain calculus using fluent and fluxions. He said that the rate of generation is the fluxion of thefluent, which is denoted by the variable with a dot over it. Then the rate of the fluxion is the second fluxion, whichhas two dots over it. These fluxions were thought of, as very close to zero but not quite zero. But when you multiplytwo fluxions together you get something that is so close to zero that it is treated as zero. Newton took derivatives byreplacing all the x values with x+ x and all the y values with y+ y and then used derivative rules to take the derivativeand solve for y

x . [8] Here is an example:x2 + y2 = 1

(x+ x)2 + (y + y)2 = 1

x2 + 2xx+ x2 + y2 + 2yy + y2 = 1

x2 + 2xx+ y2 + 2yy = 1

Using the fact that x2 + y2 = 1 we can see 2xx+ 2yy = 0 and yx = −2x

2y so yx = −x

y .Newton described mathematical quantities to be like continuous motion. This motion, he said, could be thought ofin the same way that a point traces a curve. He defined this quantity and called it a “fluent”. He went on to name therate at which these quantities change. Newton called this the “fluxion of the fluent” and he represented it by x .So, if the fluent was represented by x, Newton denoted its fluxion by x , the second fluxion by x , and so on. Thiscan be related to the modern language we use to describe derivatives. In modern language, the fluxion of the variablex relative to an independent time-variable t would be its velocity dx/dt. In other words, the derivative of f(x) withrespect to time, t, is dx/dt.

Moment of the fluent

Newton called o the moment of the fluent. The moment of the fluent represents the infinitely small part by which afluent was increased in a small time interval. Once he allowed himself to divide through by o (although o can not betreated as zero because that would make the division illegitimate). Newton decided it was justifiable to drop all termscontaining o.

2.2.4 Euler’s notation

Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df. Thesecond derivative is denoted D2f, and the nth derivative is denoted Dnf.If y = f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variablex. Euler’s notation is then written

Dxy or Dxf(x) ,

although this subscript is often omitted when the variable x is understood, for instance when this is the only variablepresent in the expression.Euler’s notation is useful for stating and solving linear differential equations.

20 CHAPTER 2. DERIVATIVE

2.3 Rules of computation

Main article: Differentiation rules

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient,and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of otherfunctions are more easily computed using rules for obtaining derivatives of more complicated functions from simplerones.

2.3.1 Rules for basic functions

Most derivative computations eventually require taking the derivative of some common functions. The followingincomplete list gives some of the most frequently used functions of a single real variable and their derivatives.

• Derivatives of powers: if

f(x) = xr,

where r is any real number, then

f ′(x) = rxr−1,

wherever this function is defined. For example, if f(x) = x1/4 , then

f ′(x) = (1/4)x−3/4,

and the derivative function is defined only for positive x, not for x = 0. When r = 0, this rule implies that f′(x) is zerofor x ≠ 0, which is almost the constant rule (stated below).

• Exponential and logarithmic functions:

d

dxex = ex.

d

dxax = ln(a)ax.

d

dxln(x) = 1

x, x > 0.

d

dxloga(x) =

1

x ln(a) .

• Trigonometric functions:

d

dxsin(x) = cos(x).

d

dxcos(x) = − sin(x).

d

dxtan(x) = sec2(x) = 1

cos2(x) = 1 + tan2(x).

• Inverse trigonometric functions:

d

dxarcsin(x) = 1√

1− x2,−1 < x < 1.

d

dxarccos(x) = − 1√

1− x2,−1 < x < 1.

d

dxarctan(x) = 1

1 + x2

2.4. DERIVATIVES IN HIGHER DIMENSIONS 21

2.3.2 Rules for combined functions

In many cases, complicated limit calculations by direct application of Newton’s difference quotient can be avoidedusing differentiation rules. Some of the most basic rules are the following.

• Constant rule: if f(x) is constant, then

f ′ = 0.

• Sum rule:

(αf + βg)′ = αf ′ + βg′ for all functions f and g and all real numbers α and β .

• Product rule:

(fg)′ = f ′g + fg′ for all functions f and g. As a special case, this rule includes the fact (αf)′ = αf ′

whenever α is a constant, because α′f = 0 · f = 0 by the constant rule.

• Quotient rule:

(fg

)′= f ′g−fg′

g2 for all functions f and g at all inputs where g ≠ 0.

• Chain rule: If f(x) = h(g(x)) , then

f ′(x) = h′(g(x)) · g′(x).

2.3.3 Computation example

The derivative of

f(x) = x4 + sin(x2)− ln(x)ex + 7

is

f ′(x) = 4x(4−1) +d(x2

)dx

cos(x2)− d (lnx)dx

ex − lnxd (ex)

dx+ 0

= 4x3 + 2x cos(x2)− 1

xex − ln(x)ex.

Here the second term was computed using the chain rule and third using the product rule. The known derivatives ofthe elementary functions x2, x4, sin(x), ln(x) and exp(x) = ex, as well as the constant 7, were also used.

2.4 Derivatives in higher dimensions

See also: Vector calculus and Multivariable calculus

22 CHAPTER 2. DERIVATIVE

2.4.1 Derivatives of vector valued functions

A vector-valued function y(t) of a real variable sends real numbers to vectors in some vector space Rn. A vector-valued function can be split up into its coordinate functions y1(t), y2(t), …, yn(t), meaning that y(t) = (y1(t), ..., yn(t)).This includes, for example, parametric curves in R2 or R3. The coordinate functions are real valued functions, sothe above definition of derivative applies to them. The derivative of y(t) is defined to be the vector, called the tangentvector, whose coordinates are the derivatives of the coordinate functions. That is,

y′(t) = (y′1(t), . . . , y′n(t)).

Equivalently,

y′(t) = limh→0

y(t+ h)− y(t)h

,

if the limit exists. The subtraction in the numerator is subtraction of vectors, not scalars. If the derivative of y existsfor every value of t, then y′ is another vector valued function.If e1, …, en is the standard basis for Rn, then y(t) can also be written as y1(t)e1 + … + yn(t)en. If we assume thatthe derivative of a vector-valued function retains the linearity property, then the derivative of y(t) must be

y′1(t)e1 + · · ·+ y′n(t)en

because each of the basis vectors is a constant.This generalization is useful, for example, if y(t) is the position vector of a particle at time t; then the derivative y′(t)is the velocity vector of the particle at time t.

2.4.2 Partial derivatives

Main article: Partial derivative

Suppose that f is a function that depends on more than one variable—for instance,

f(x, y) = x2 + xy + y2.

f can be reinterpreted as a family of functions of one variable indexed by the other variables:

f(x, y) = fx(y) = x2 + xy + y2.

In other words, every value of x chooses a function, denoted fx, which is a function of one real number.[9] That is,

x 7→ fx,

fx(y) = x2 + xy + y2.

Once a value of x is chosen, say a, then f(x, y) determines a function fa that sends y to a2 + ay + y2:

fa(y) = a2 + ay + y2.

In this expression, a is a constant, not a variable, so fa is a function of only one real variable. Consequently, thedefinition of the derivative for a function of one variable applies:

2.4. DERIVATIVES IN HIGHER DIMENSIONS 23

f ′a(y) = a+ 2y.

The above procedure can be performed for any choice of a. Assembling the derivatives together into a function givesa function that describes the variation of f in the y direction:

∂f

∂y(x, y) = x+ 2y.

This is the partial derivative of f with respect to y. Here ∂ is a rounded d called the partial derivative symbol. Todistinguish it from the letter d, ∂ is sometimes pronounced “der”, “del”, or “partial” instead of “dee”.In general, the partial derivative of a function f(x1, …, xn) in the direction xi at the point (a1 …, an) is defined tobe:

∂f

∂xi(a1, . . . , an) = lim

h→0

f(a1, . . . , ai + h, . . . , an)− f(a1, . . . , ai, . . . , an)

h.

In the above difference quotient, all the variables except xi are held fixed. That choice of fixed values determines afunction of one variable

fa1,...,ai−1,ai+1,...,an(xi) = f(a1, . . . , ai−1, xi, ai+1, . . . , an),

and, by definition,

dfa1,...,ai−1,ai+1,...,an

dxi(ai) =

∂f

∂xi(a1, . . . , an).

In other words, the different choices of a index a family of one-variable functions just as in the example above.This expression also shows that the computation of partial derivatives reduces to the computation of one-variablederivatives.An important example of a function of several variables is the case of a scalar-valued function f(x1, ..., xn) on adomain in Euclidean space Rn (e.g., on R2 or R3). In this case f has a partial derivative ∂f/∂xj with respect to eachvariable xj. At the point a, these partial derivatives define the vector

∇f(a) =

(∂f

∂x1(a), . . . ,

∂f

∂xn(a)

).

This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is avector-valued function ∇f that takes the point a to the vector ∇f(a). Consequently, the gradient determines a vectorfield.

2.4.3 Directional derivatives

Main article: Directional derivative

If f is a real-valued function on Rn, then the partial derivatives of f measure its variation in the direction of thecoordinate axes. For example, if f is a function of x and y, then its partial derivatives measure the variation in f inthe x direction and the y direction. They do not, however, directly measure the variation of f in any other direction,such as along the diagonal line y = x. These are measured using directional derivatives. Choose a vector

v = (v1, . . . , vn).

24 CHAPTER 2. DERIVATIVE

The directional derivative of f in the direction of v at the point x is the limit

Dvf(x) = limh→0

f(x+ hv)− f(x)h

.

In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector.Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector.To see how this works, suppose that v = λu. Substitute h = k/λ into the difference quotient. The difference quotientbecomes:

f(x+ (k/λ)(λu))− f(x)k/λ

= λ · f(x+ ku)− f(x)k

.

This is λ times the difference quotient for the directional derivative of f with respect to u. Furthermore, taking thelimit as h tends to zero is the same as taking the limit as k tends to zero because h and k are multiples of each other.Therefore, Dᵥ(f) = λDᵤ(f). Because of this rescaling property, directional derivatives are frequently considered onlyfor unit vectors.If all the partial derivatives of f exist and are continuous at x, then they determine the directional derivative of f inthe direction v by the formula:

Dvf(x) =n∑

j=1

vj∂f

∂xj.

This is a consequence of the definition of the total derivative. It follows that the directional derivative is linear in v,meaning that Dᵥ ₊ (f) = Dᵥ(f) + D (f).The same definition also works when f is a function with values in Rm. The above definition is applied to eachcomponent of the vectors. In this case, the directional derivative is a vector in Rm.

2.4.4 Total derivative, total differential and Jacobian matrix

Main article: Total derivative

When f is a function from an open subset of Rn to Rm, then the directional derivative of f in a chosen direction is thebest linear approximation to f at that point and in that direction. But when n > 1, no single directional derivative cangive a complete picture of the behavior of f. The total derivative gives a complete picture by considering all directionsat once. That is, for any vector v starting at a, the linear approximation formula holds:

f(a+ v) ≈ f(a) + f ′(a)v.

Just like the single-variable derivative, f ′(a) is chosen so that the error in this approximation is as small as possible.If n and m are both one, then the derivative f ′(a) is a number and the expression f ′(a)v is the product of two numbers.But in higher dimensions, it is impossible for f ′(a) to be a number. If it were a number, then f ′(a)v would be avector in Rn while the other terms would be vectors in Rm, and therefore the formula would not make sense. For thelinear approximation formula to make sense, f ′(a) must be a function that sends vectors in Rn to vectors in Rm, andf ′(a)v must denote this function evaluated at v.To determine what kind of function it is, notice that the linear approximation formula can be rewritten as

f(a+ v)− f(a) ≈ f ′(a)v.

Notice that if we choose another vector w, then this approximate equation determines another approximate equationby substituting w for v. It determines a third approximate equation by substituting both w for v and a + v for a. Bysubtracting these two new equations, we get

2.4. DERIVATIVES IN HIGHER DIMENSIONS 25

f(a+ v+ w)− f(a+ v)− f(a+ w) + f(a) ≈ f ′(a+ v)w− f ′(a)w.

If we assume that v is small and that the derivative varies continuously in a, then f ′(a + v) is approximately equal tof ′(a), and therefore the right-hand side is approximately zero. The left-hand side can be rewritten in a different wayusing the linear approximation formula with v + w substituted for v. The linear approximation formula implies:

0 ≈ f(a+ v+ w)− f(a+ v)− f(a+ w) + f(a)= (f(a+ v+ w)− f(a))− (f(a+ v)− f(a))− (f(a+ w)− f(a))≈ f ′(a)(v+ w)− f ′(a)v− f ′(a)w.

This suggests that f ′(a) is a linear transformation from the vector space Rn to the vector space Rm. In fact, it ispossible to make this a precise derivation by measuring the error in the approximations. Assume that the error inthese linear approximation formula is bounded by a constant times ||v||, where the constant is independent of v butdepends continuously on a. Then, after adding an appropriate error term, all of the above approximate equalities canbe rephrased as inequalities. In particular, f ′(a) is a linear transformation up to a small error term. In the limit as vand w tend to zero, it must therefore be a linear transformation. Since we define the total derivative by taking a limitas v goes to zero, f ′(a) must be a linear transformation.In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limitof difference quotients. However, the usual difference quotient does not make sense in higher dimensions because itis not usually possible to divide vectors. In particular, the numerator and denominator of the difference quotient arenot even in the same vector space: The numerator lies in the codomain Rm while the denominator lies in the domainRn. Furthermore, the derivative is a linear transformation, a different type of object from both the numerator anddenominator. To make precise the idea that f ′(a) is the best linear approximation, it is necessary to adapt a differentformula for the one-variable derivative in which these problems disappear. If f : R → R, then the usual definition ofthe derivative may be manipulated to show that the derivative of f at a is the unique number f ′(a) such that

limh→0

f(a+ h)− f(a)− f ′(a)h

h= 0.

This is equivalent to

limh→0

|f(a+ h)− f(a)− f ′(a)h||h|

= 0

because the limit of a function tends to zero if and only if the limit of the absolute value of the function tends to zero.This last formula can be adapted to the many-variable situation by replacing the absolute values with norms.The definition of the total derivative of f at a, therefore, is that it is the unique linear transformation f ′(a) : Rn →Rm such that

limh→0

∥f(a+ h)− f(a)− f ′(a)h∥∥h∥ = 0.

Here h is a vector in Rn, so the norm in the denominator is the standard length on Rn. However, f′(a)h is a vector inRm, and the norm in the numerator is the standard length on Rm. If v is a vector starting at a, then f ′(a)v is calledthe pushforward of v by f and is sometimes written f∗v.If the total derivative exists at a, then all the partial derivatives and directional derivatives of f exist at a, and for allv, f ′(a)v is the directional derivative of f in the direction v. If we write f using coordinate functions, so that f = (f1,f2, ..., fm), then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is calledthe Jacobian matrix of f at a:

f ′(a) = Jaca =(∂fi∂xj

)ij

.

26 CHAPTER 2. DERIVATIVE

The existence of the total derivative f′(a) is strictly stronger than the existence of all the partial derivatives, but ifthe partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and dependscontinuously on a.The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if f is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. The Jacobianmatrix reduces to a 1×1 matrix whose only entry is the derivative f′(x). This 1×1 matrix satisfies the property thatf(a + h) − f(a) − f ′(a)h is approximately zero, in other words that

f(a+ h) ≈ f(a) + f ′(a)h.

Up to changing variables, this is the statement that the function x 7→ f(a) + f ′(a)(x− a) is the best linear approx-imation to f at a.The total derivative of a function does not give another function in the same way as the one-variable case. This isbecause the total derivative of a multivariable function has to record much more information than the derivative ofa single-variable function. Instead, the total derivative gives a function from the tangent bundle of the source to thetangent bundle of the target.The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a functionon the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative,called a jet, cannot be a linear transformation because higher-order derivatives reflect subtle geometric information,such as concavity, which cannot be described in terms of linear data such as vectors. It cannot be a function onthe tangent bundle because the tangent bundle only has room for the base space and the directional derivatives.Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the jet bundle. Therelation between the total derivative and the partial derivatives of a function is paralleled in the relation between thekth order jet of a function and its partial derivatives of order less than or equal to k.By repeatedly taking the total derivative, one obtains higher versions of the Fréchet derivative, specialized to Rp. Thekth order total derivative may be interpreted as a map

Dkf : Rn → Lk(Rn × · · · × Rn,Rm)

which takes a point x in Rn and assigns to it an element of the space of k-linear maps from Rn to Rm – the “best” (ina certain precise sense) k-linear approximation to f at that point. By precomposing it with the diagonal map Δ, x →(x, x), a generalized Taylor series may be begun as

f(x) ≈ f(a) + (Df)(x) + (D2f)(∆(x− a)) + · · ·= f(a) + (Df)(x− a) + (D2f)(x− a, x− a) + · · ·

= f(a) +∑i

(Df)i(x− a)i +∑j,k

(D2f)jk(x− a)j(x− a)k + · · ·

where f(a) is identified with a constant function, (x − a)i are the components of the vector x − a, and (D f)i and (D2

f)j k are the components of D f and D2 f as linear transformations.

2.5 Generalizations

Main article: Derivative (generalizations)

The concept of a derivative can be extended to many other settings. The common thread is that the derivative of afunction at a point serves as a linear approximation of the function at that point.

• An important generalization of the derivative concerns complex functions of complex variables, such as func-tions from (a domain in) the complex numbers C to C. The notion of the derivative of such a function is

2.6. HISTORY 27

obtained by replacing real variables with complex variables in the definition. If C is identified with R2 bywriting a complex number z as x + i y, then a differentiable function from C to C is certainly differentiableas a function from R2 to R2 (in the sense that its partial derivatives all exist), but the converse is not true ingeneral: the complex derivative only exists if the real derivative is complex linear and this imposes relationsbetween the partial derivatives called the Cauchy–Riemann equations – see holomorphic functions.

• Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speakingsuch a manifold M is a space that can be approximated near each point x by a vector space called its tangentspace: the prototypical example is a smooth surface in R3. The derivative (or differential) of a (differentiable)map f: M → N between manifolds, at a point x in M, is then a linear map from the tangent space of M at x tothe tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M andN. This definition is fundamental in differential geometry and has many uses – see pushforward (differential)and pullback (differential geometry).

• Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spacesand Fréchet spaces. There is a generalization both of the directional derivative, called the Gâteaux derivative,and of the differential, called the Fréchet derivative.

• One deficiency of the classical derivative is that not very many functions are differentiable. Nevertheless, thereis a way of extending the notion of the derivative so that all continuous functions and many other functions canbe differentiated using a concept known as the weak derivative. The idea is to embed the continuous functionsin a larger space called the space of distributions and only require that a function is differentiable “on average”.

• The properties of the derivative have inspired the introduction and study of many similar objects in algebra andtopology — see, for example, differential algebra.

• The discrete equivalent of differentiation is finite differences. The study of differential calculus is unified withthe calculus of finite differences in time scale calculus.

• Also see arithmetic derivative.

2.6 History

Main article: History of calculus

2.7 See also• Applications of derivatives

• Automatic differentiation

• Differentiability class

• Differentiation rules

• Differintegral

• Fractal derivative

• Generalizations of the derivative

• Hasse derivative

• History of calculus

• Integral

• Linearization

• Mathematical analysis

28 CHAPTER 2. DERIVATIVE

• Multiplicative inverse

• Numerical differentiation

• Radon–Nikodym theorem

• Symmetric derivative

2.8 Notes[1] Differential calculus, as discussed in this article, is a very well established mathematical discipline for which there are many

sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.

[2] Spivak 1994, chapter 10.

[3] Despite this, it is still possible to take the derivative in the sense of distributions. The result is nine times the Dirac measurecentered at a.

[4] Banach, S. (1931), “Uber die Baire’sche Kategorie gewisser Funktionenmengen”, Studia. Math. (3): 174–179.. Cited byHewitt, E and Stromberg, K (1963), Real and abstract analysis, Springer-Verlag, Theorem 17.8

[5] Apostol 1967, §4.18

[6] In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors. Someauthors do not assign a meaning to du by itself, but only as part of the symbol du/dx. Others define dx as an independentvariable, and define du by du = dx·f′(x). In non-standard analysis du is defined as an infinitesimal. It is also interpreted asthe exterior derivative of a function u. See differential (infinitesimal) for further information.

[7] “The Notation of Differentiation”. MIT. 1998. Retrieved 24 October 2012.

[8] Burton, D. (2011). Gottfried Leibniz: The Calculus Controversy. The History of Mathematics: An Introduction (). NewYork, NY: McGraw-Hill.

[9] This can also be expressed as the adjointness between the product space and function space constructions.

2.9 References

2.9.1 Print

• Anton, Howard; Bivens, Irl; Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single andMultivariable (8th ed.), New York: Wiley, ISBN 978-0-471-47244-5

• Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1

• Apostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications1 (2nd ed.), Wiley, ISBN 978-0-471-00007-5

• Courant, Richard; John, Fritz (December 22, 1998), Introduction to Calculus and Analysis, Vol. 1, Springer-Verlag, ISBN 978-3-540-65058-4

• Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN978-0-03-029558-4

• Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006), Calculus: Early TranscendentalFunctions (4th ed.), Houghton Mifflin Company, ISBN 978-0-618-60624-5

• Spivak, Michael (September 1994), Calculus (3rd ed.), Publish or Perish, ISBN 978-0-914098-89-8

• Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7

• Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), NewYork: St. Martin’s Press, ISBN 978-0-312-18548-0

2.9. REFERENCES 29

2.9.2 Online books

• Crowell, Benjamin (2003), Calculus

• (Govt. of TN), TamilNadu Textbook Corporation (2006), Mathematics- vol.2 (PDF)

• Garrett, Paul (2004), Notes on First-Year Calculus, University of Minnesota

• Hussain, Faraz (2006), Understanding Calculus

• Keisler, H. Jerome (2000), Elementary Calculus: An Approach Using Infinitesimals

• Mauch, Sean (2004), Unabridged Version of Sean’s Applied Math Book

• Sloughter, Dan (2000), Difference Equations to Differential Equations

• Strang, Gilbert (1991), Calculus

• Stroyan, Keith D. (1997), A Brief Introduction to Infinitesimal Calculus

• Wikibooks, Calculus

2.9.3 Web pages

• Hazewinkel, Michiel, ed. (2001), “Derivative”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Khan Academy: “Newton, Leibniz, and Usain Bolt”

• Weisstein, Eric W. "Derivative." From MathWorld

• Derivatives of Trigonometric functions, UBC

Chapter 3

Tangent

For the tangent function, see Trigonometric functions#Sine, cosine and tangent. For other uses, see Tangent (disam-biguation).

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that “just

Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.

touches” the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve.[1]

More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passesthrough the point (c, f (c)) on the curve and has slope f ' (c) where f ' is the derivative of f. A similar definitionapplies to space curves and curves in n-dimensional Euclidean space.As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangentline is “going in the same direction” as the curve, and is thus the best straight-line approximation to the curve at thatpoint.Similarly, the tangent plane to a surface at a given point is the plane that “just touches” the surface at that point.The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensivelygeneralized; see Tangent space.

30

3.1. HISTORY 31

Tangent plane to a sphere

The word tangent comes from the Latin tangere, to touch.

3.1 History

Euclid makes several references to the tangent (ἐφαπτομένη) to a circle in book III of the Elements (c. 300 BC).[2]

In Apollonius work Conics (ca. 225 BC) he defines a tangent as being a line such that no other straight line could fallbetween it and the curve.[3] Archimedes (ca. 287–222 BC) found the tangent to an Archimedean spiral by consideringthe path of a point moving along the curve.[3]

In the 1630s Fermat developed the technique of adequality to calculate tangents and other problems in analysis andused this to calculate tangents to the parabola, the technique of adeqality is similar to taking the difference betweenf(x+h) and f(x) and dividing by a power of h . Independently Descartes used his method of normals based on theobservation that the radius of a circle is always normal to the circle itself.[4]

These methods led to the development of differential calculus in the 17th Century. Many people contributed, Robervaldiscovered a general method of drawing tangents, by considering a curve as described by a moving point whose motionis the resultant of several simpler motions.[5] René-François de Sluse and Johannes Hudde found algebraic algorithmsfor finding tangents.[6] Further developments we due to John Wallis, Isaac Barrow until the full theory of Isaac Newtonand Gottfried Leibniz.An 1828 definition of a tangent was “a right line which touches a curve, but which when produced, does not cutit”.[7] This old definition prevents inflection points from having any tangent. It has been dismissed and the moderndefinitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely closepoints on the curve.

3.2 Tangent line to a curve

The intuitive notion that a tangent line “touches” a curve can be made more explicit by considering the sequence ofstraight lines (secant lines) passing through two points, A and B, those that lie on the function curve. The tangent atA is the limit when point B approximates or tends to A. The existence and uniqueness of the tangent line depends on

32 CHAPTER 3. TANGENT

Diameter

Chord

Tang

entSecant

Radius

A tangent, a chord, and a secant to a circle

a certain type of mathematical smoothness, known as “differentiability.” For example, if two circular arcs meet at asharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression ofsecant lines depends on the direction in which “point B" approaches the vertex.At most points, the tangent touches the curve without crossing it (though it may, when continued, cross the curve atother places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called aninflection point. Circles, parabolas, hyperbolas and ellipses do not have any inflection point, but more complicatedcurves do have, like the graph of a cubic function, which has exactly one inflection point.Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, andyet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of atriangle and not intersecting the triangle—where the tangent line does not exist for the reasons explained above. Inconvex geometry, such lines are called supporting lines.

3.2.1 Analytical approach

The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methodsthat are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent lineproblem, was one of the central questions leading to the development of calculus in the 17th century. In the second

3.2. TANGENT LINE TO A CURVE 33

At each point, the moving line is always tangent to the curve. Its slope is the derivative; green marks positive derivative, red marksnegative derivative and black marks zero derivative.

book of his Geometry, René Descartes[8] said of the problem of constructing the tangent to a curve, “And I dare saythat this is not only the most useful and most general problem in geometry that I know, but even that I have everdesired to know”.[9]

Intuitive description

Suppose that a curve is given as the graph of a function, y = f(x). To find the tangent line at the point p = (a, f(a)),consider another nearby point q = (a + h, f(a + h)) on the curve. The slope of the secant line passing through p andq is equal to the difference quotient

f(a+ h)− f(a)

h.

As the point q approaches p, which corresponds to making h smaller and smaller, the difference quotient shouldapproach a certain limiting value k, which is the slope of the tangent line at the point p. If k is known, the equationof the tangent line can be found in the point-slope form:

34 CHAPTER 3. TANGENT

y − f(a) = k(x− a).

More rigorous description

To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching acertain limiting value k. The precise mathematical formulation was given by Cauchy in the 19th century and is basedon the notion of limit. Suppose that the graph does not have a break or a sharp edge at p and it is neither plumb nortoo wiggly near p. Then there is a unique value of k such that, as h approaches 0, the difference quotient gets closerand closer to k, and the distance between them becomes negligible compared with the size of h, if h is small enough.This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for thefunction f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation ofthe tangent line can be stated as follows:

y = f(a) + f ′(a)(x− a).

Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the powerfunction, trigonometric functions, exponential function, logarithm, and their various combinations. Thus, equationsof the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.

How the method can fail

Calculus also demonstrates that there are functions and points on their graphs for which the limit determining theslope of the tangent line does not exist. For these points the function f is non-differentiable. There are two possiblereasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangentexists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or thegraph exhibits one of three behaviors that precludes a geometric tangent.The graph y = x1/3 illustrates the first possibility: here the difference quotient at a = 0 is equal to h1/3/h = h−2/3, whichbecomes very large as h approaches 0. This curve has a tangent line at the origin that is vertical.The graph y = x2/3 illustrates another possibility: this graph has a cusp at the origin. This means that, when happroaches 0, the difference quotient at a = 0 approaches plus or minus infinity depending on the sign of x. Thusboth branches of the curve are near to the half vertical line for which y=0, but none is near to the negative part ofthis line. Basically, there is no tangent at the origin in this case, but in some context one may consider this line as atangent, and even, in algebraic geometry, as a double tangent.The graph y = |x| of the absolute value function consists of two straight lines with different slopes joined at the origin.As a point q approaches the origin from the right, the secant line always has slope 1. As a point q approaches theorigin from the left, the secant line always has slope −1. Therefore, there is no unique tangent to the graph at theorigin. Having two different (but finite) slopes is called a corner.Finally, since differentiability implies continuity, the contrapositive states discontinuity implies non-differentiability.Any such jump or point discontinuity will have no tangent line. This includes cases where one slope approachespositive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity

3.2.2 Equations

When the curve is given by y = f(x) then the slope of the tangent is dydx , so by the point–slope formula the equation

of the tangent line at (X, Y) is

y − Y =dy

dx(X) · (x−X)

where (x, y) are the coordinates of any point on the tangent line, and where the derivative is evaluated at x = X .[10]

When the curve is given by y = f(x), the tangent line’s equation can also be found[11] by using polynomial division todivide f (x) by (x−X)2 ; if the remainder is denoted by g(x) , then the equation of the tangent line is given by

3.2. TANGENT LINE TO A CURVE 35

y = g(x).

When the equation of the curve is given in the form f(x, y) = 0 then the value of the slope can be found by implicitdifferentiation, giving

dy

dx= −

∂f∂x∂f∂y

.

The equation of the tangent line at a point (X,Y) such that f(X,Y) = 0 is then[10]

∂f

∂x(X,Y ) · (x−X) +

∂f

∂y(X,Y ) · (y − Y ) = 0.

This equation remains true if ∂f∂y (X,Y ) = 0 but ∂f

∂x (X,Y ) = 0 (in this case the slope of the tangent is infinite). If∂f∂y (X,Y ) = ∂f

∂x (X,Y ) = 0, the tangent line is not defined and the point (X,Y) is said singular.For algebraic curves, computations may be simplified somewhat by converting to homogeneous coordinates. Specif-ically, let the homogeneous equation of the curve be g(x, y, z) = 0 where g is a homogeneous function of degree n.Then, if (X, Y, Z) lies on the curve, Euler’s theorem implies

∂g

∂x·X +

∂g

∂y· Y +

∂g

∂z· Z = ng(X,Y, Z) = 0.

It follows that the homogeneous equation of the tangent line is

∂g

∂x(X,Y, Z) · x+

∂g

∂y(X,Y, Z) · y + ∂g

∂z(X,Y, Z) · z = 0.

The equation of the tangent line in Cartesian coordinates can be found by setting z=1 in this equation.[12]

To apply this to algebraic curves, write f(x, y) as

f = un + un−1 + · · ·+ u1 + u0

where each ur is the sum of all terms of degree r. The homogeneous equation of the curve is then

g = un + un−1z + · · ·+ u1zn−1 + u0z

n = 0.

Applying the equation above and setting z=1 produces

∂f

∂x(X,Y ) · x+

∂f

∂y(X,Y ) · y + ∂g

∂z(X,Y, 1) = 0

as the equation of the tangent line.[13] The equation in this form is often simpler to use in practice since no furthersimplification is needed after it is applied.[12]

If the curve is given parametrically by

x = x(t), y = y(t)

then the slope of the tangent is

36 CHAPTER 3. TANGENT

dy

dx=

dydtdxdt

giving the equation for the tangent line at t = T, X = x(T ), Y = y(T ) as[14]

dx

dt(T ) · (y − Y ) =

dy

dt(T ) · (x−X).

If dxdt (T ) =

dydt (T ) = 0, , the tangent line is not defined. However, it may occur that the tangent line exists and may

be computed from an implicit equation of the curve.

3.2.3 Normal line to a curve

The line perpendicular to the tangent line to a curve at the point of tangency is called the normal line to the curve atthat point. The slopes of perpendicular lines have product −1, so if the equation of the curve is y = f(x) then slopeof the normal line is

− 1dydx

and it follows that the equation of the normal line at (X, Y) is

(x−X) +dy

dx(y − Y ) = 0.

Similarly, if the equation of the curve has the form f(x, y) = 0 then the equation of the normal line is given by[15]

∂f

∂y(x−X)− ∂f

∂x(y − Y ) = 0.

If the curve is given parametrically by

x = x(t), y = y(t)

then the equation of the normal line is[14]

dx

dt(x−X) +

dy

dt(y − Y ) = 0.

3.2.4 Angle between curves

See also: Angle § Angle between curves

The angle between two curves at a point where they intersect is defined as the angle between their tangent lines atthat point. More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, andorthogonal if their tangent lines are orthogonal.[16]

3.2. TANGENT LINE TO A CURVE 37

a 3a

The limaçon trisectrix: a curve with two tangents at the origin.

3.2.5 Multiple tangents at a point

The formulas above fail when the point is a singular point. In this case there may be two or more branches of the curvewhich pass through the point, each branch having its own tangent line. When the point is the origin, the equationsof these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowestdegree terms from the original equation. Since any point can be made the origin by a change of variables, this givesa method for finding the tangent lines at any singular point.For example, the equation of the limaçon trisectrix shown to the right is

(x2 + y2 − 2ax)2 = a2(x2 + y2).

Expanding this and eliminating all but terms of degree 2 gives

a2(3x2 − y2) = 0

which, when factored, becomes

y = ±√3x.

So these are the equations of the two tangent lines through the origin.[17]

When the curve is not self-crossing, the tangent at a reference point may still not be uniquely defined because thecurve is not differentiable at that point although it is differentiable elsewhere. In this case the left and right derivatives

38 CHAPTER 3. TANGENT

are defined as the limits of the derivative as the point at which it is evaluated approaches the reference point fromrespectively the left (lower values) or the right (higher values). For example, the curve y = |x | is not differentiable atx = 0: its left and right derivatives have respective slopes –1 and 1; the tangents at that point with those slopes arecalled the left and right tangents.[18]

Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, forexample, for the curve y = x 2/3, for which both the left and right derivatives at x = 0 are infinite; both the left andright tangent lines have equation x = 0.

3.3 Tangent circles

Main article: Tangent circlesTwo circles of non-equal radius, both in the same plane, are said to be tangent to each other if they meet at only onepoint. Equivalently, two circles, with radii of ri and centers at (xi, yi), for i = 1, 2 are said to be tangent to each otherif

(x1 − x2)2+ (y1 − y2)

2= (r1 ± r2)

2.

• Two circles are externally tangent if the distance between their centres is equal to the sum of their radii.

(x1 − x2)2+ (y1 − y2)

2= (r1 + r2)

2.

• Two circles are internally tangent if the distance between their centres is equal to the difference between theirradii.[19]

(x1 − x2)2+ (y1 − y2)

2= (r1 − r2)

2.

3.4 Surfaces and higher-dimensional manifolds

Main article: Tangent space

The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case ofcurves. It is the best approximation of the surface by a plane at p, and can be obtained as the limiting position of theplanes passing through 3 distinct points on the surface close to p as these points converge to p. More generally, thereis a k-dimensional tangent space at each point of a k-dimensional manifold in the n-dimensional Euclidean space.

3.5 See also• Newton’s method

• Normal (geometry)

• Osculating circle

• Osculating curve

• Perpendicular

• Subtangent

• Supporting line

• Tangent cone

• Tangential angle

3.5. SEE ALSO 39

Two pairs of tangent circles. Above internally and below externally tangent

• Tangential component

• Tangent lines to circles

• Multiplicity (mathematics)#Behavior of a polynomial function near a multiple root

• Algebraic curve#Tangent at a point

40 CHAPTER 3. TANGENT

3.6 References[1] Leibniz, G., Nova methodus pro maximis et minimis ..., Acta Erud., Oct. 1684

[2] Euclid. “Euclid’s Elements”. Retrieved 1 June 2015.

[3] Shenk, Al. “e-CALCULUS Section 2.8” (PDF). p. 2.8. Retrieved 1 June 2015.

[4] Katz, Victor J. (2008). A History of Mathematics (3rd ed.). Addison Wesley. p. 510. ISBN 978-0321387004.

[5] Wolfson, Paul R. (2001). “The Crooked Made Straight: Roberval and Newton on Tangents”. The American MathematicalMonthly 108 (3): 206–216.

[6] Katz, Victor J. (2008). A History of Mathematics (3rd ed.). Addison Wesley. pp. 512–514. ISBN 978-0321387004.

[7] Noah Webster, American Dictionary of the English Language (New York: S. Converse, 1828), vol. 2, p.733,

[8] Descartes, René (1954). The geometry of René Descartes. Courier Dover. p. 95. ISBN 0-486-60068-8.

[9] R. E. Langer (October 1937). “Rene Descartes”. American Mathematical Monthly (Mathematical Association of America)44 (8): 495–512. doi:10.2307/2301226. JSTOR 2301226.

[10] Edwards Art. 191

[11] Strickland-Constable, Charles, “A simple method for finding tangents to polynomial graphs”, Mathematical Gazette, Novem-ber 2005, 466–467.

[12] Edwards Art. 192

[13] Edwards Art. 193

[14] Edwards Art. 196

[15] Edwards Art. 194

[16] Edwards Art. 195

[17] Edwards Art. 197

[18] Thomas, George B. Jr., and Finney, Ross L. (1979), Calculus and Analytic Geometry, Addison Wesley Publ. Co.: p. 140.

[19] Circles For Leaving Certificate Honours Mathematics by Thomas O’Sullivan 1997

• J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 143 ff.

3.7 External links• Hazewinkel, Michiel, ed. (2001), “Tangent line”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

• Weisstein, Eric W., “Tangent Line”, MathWorld.

• Tangent to a circle With interactive animation

• Tangent and first derivative — An interactive simulation

• The Tangent Parabola by John H. Mathews

3.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 41

3.8 Text and image sources, contributors, and licenses

3.8.1 Text• Curve Source: https://en.wikipedia.org/wiki/Curve?oldid=680215356 Contributors: Zundark, XJaM, Tedernst, Mei~enwiki, Hfastedge,

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42 CHAPTER 3. TANGENT

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