the derivative 3.1 introduction to the derivativestaff 150a/chapter 3 lecture notes.pdf · for all...

3The Derivative

3.1 Introduction to the Derivative

Consider a function f and a line that passes through the points (c, f(c)) and (c + ∆x, f(c + ∆x)).y

x

Definition. If f is defined on an open interval containing c, and if

lim∆x→0

∆y

∆x=

exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of fat the point (c, f(c)).

Example 1. Find the slope of the graph of y = 3x + 2 at the point (1, 5).

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3 The Derivative

Example 2. Find the slope of the tangent lines to the graph of f(x) = x2 − 1 at the points (0,−1)and (1, 0).

Definition. The derivative of f at x is given by

f ′(x) =

provided the limit exists. For all x for which this limit exists, f ′ is a function of x.

Example 3. Find the derivative of f(x) = x2 + 3x + 2.

2


Example 4. Find f ′(x) for f(x) = 1x. Use your result to find the slope of the graph of f(x) at the

points (12, 2), (1, 1), and (2, 1

2)

Example 5. Find the derivative with respect to t if y =√

t.

3

3 The Derivative

Notation. We may write the derivative with respect to x in the following ways:

• f ′(x)

• dy

dx

• y′

• d

dx

[

f(x)]

• Dx[y]

The Alternate Form of the Derivative

The following form of the derivative is useful for investigating the relationship between differentia-bility and continuity.

f ′(c) =

y

x

Example 6. Use the alternate form of the derivative to find f ′(c) if f(x) = x2.

4


Example 7. Use the alternate form of the derivative to investigate the differentiability at x = 0 forf(x) = |x|.

Example 8. Use the alternate form of the derivative to investigate the differentiability at x = 0 forf(x) = x1/3.

5

3 The Derivative

Theorem 3.1. If f is differentiable at x = c, then f is continuous at x = c.

6

3.2 Basic Differentiation


Theorem 3.2. If k is a real number, then

d

dx[k] =

y

x

Recall

From the binomial theorem that

(x + ∆x)n = xn + nxn−1(∆x) +n(n − 1)xn−2

2(∆x)2 + · · · + (∆x)n

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3 The Derivative

Theorem 3.3 (The Power Rule). If n is a rational number, then the function f(x) = xn is

differentiable and

d

dx[xn] =

For f to be differentiable at x = 0, n must be a number such that xn−1 is defined on an openinterval containing 0.

Example 1. Find the derivativeof f(x) = x5.

Example 2. Find the derivativeof g(x) = 4

√x.

Example 3. Find the derivativeof h(x) = 1

x3 .

8


Example 4. Find the slope of the graph of f(x) = x6 when x = −1, 0, and 1.

Example 5. Find an equation of the tangent line to the graph of f(x) = x3 when x = 2.

9

3 The Derivative

Theorem 3.4. If f is differentiable and k is a real number, then kf is differentiable and

d

dx[kf(x)] =

Example 6. Findd

dx

[

3

x

]

. Example 7. Findd

dt

[

2

3t2

]

.

10


Example 8. Findd

dx[3√

x]. Example 9. Findd

dx

[

1

54√

x3

]

.

Theorem 3.5. Let f and g be differentiable functions. Then, f ± g is differentiable and

d

dx

[

f(x) ± g(x)]

=

Example 10. Findd

dx[x2 − 3x + 2].

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3 The Derivative

Theorem 3.6.

1.d

dx[sin x] = 2.

d

dx[cos x] =

Example 11. Findd

dx[√

2 sin x]. Example 12. Findd

dx[x2 + cos x].

12


Theorem 3.7.

d

dx[ex] =

Example 13. Findd

dx[x3 + ex]. Example 14. Find

d

dx[cos x + ex].

13

3 The Derivative

Let s(t) be a position function with respect to time t. Then, the average velocity is given by

average velocity =

Example 15. If an object is dropped from a height of 50 feet, the height s of the object at time tis given by

s(t) = −16t2 + 50

where s is measured in feet and t is measured in seconds. Find the average velocity over each of thefollowing time intervals: [1, 2], [1, 1.5], and [1, 1.1].

The velocity of an object at time t is given by

v(t) =

where

s(t) =1

2gt2 + v0t + s0

is the position of the object at time t having the acceleration due to gravity g = −32 feet perseconds squared, v0 and s0 are the initial velocity and initial position respectively.

14


Example 16. At the time t = 0, an object is propelled upward from a height of 80 feet. The positionof the object is given by

s(t) = −16t2 + 64t + 80

where s is in feet and t is in seconds. When does the object hit the ground? What is the velocityof the object at impact?

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3 The Derivative

3.3 Differentiation Rules

In this section, we point out two facts:

• d

dx[f(x)g(x)] 6= d

dx[f(x)]

d

dx[g(x)] • d

dx[f(x) ÷ g(x)] 6= d

dx[f(x)] ÷ d

dx[g(x)]

That is, differentiation does not behave like the limit with respect to multiplication anddivision.

Theorem 3.8 (The Product Rule). Let f and g be differentiable functions. Then, fg is differen-

tiable and

d

dx

[

f(x)g(x)]

=

16


Example 1. Findd

dx[(3x2 − 4)(2x − 7)].

Example 2. Findd

dx[x2ex].

Example 3. Findd

dx[2x sin x + 2 cos x].

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3 The Derivative

Note. The product rule can be generalized to products of more than two factors. For example, forthree factors we have

d

dx[uvw] =

du

dxvw + u

dv

dxw + uv

dw

dx

Example 4. Findd

dx[x2 cos x sin x].

Next, we encounter the Quotient Rule and the Reciprocal Rule for derivatives. The ProductRule can be used to derive weak version of these rules. It is a “weak” version in that it does notprove that the quotient is differentiable, but only says what its derivative is if it is differentiable.

Theorem 3.9 (The Quotient Rule). Let f and g be differentiable functions. Then, f/g is differ-

entiable and

d

dx

[

f(x)

g(x)

]

=

and if f(x) = 1, we obtain the Reciprocal Rule

d

dx

[

1

g(x)

]

=

The Quotient Rule often memorized as a rhyme type song. “lo-dee-hi less hi-dee-lo, draw theline and square below”; Lo being the denominator, Hi being the numerator and “dee” being thederivative.

18


Example 5. Findd

dx

[

x − 4

x2 − 1

]

.

Example 6. Findd

dx

1 − 1

xx + 1

.

Example 7. Findd

dx

[

1

x − sin x

]

.

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3 The Derivative

Theorem 3.10.

1.d

dx[tan x] = 2.

d

dx[cot x] =

3.d

dx[sec x] = 4.

d

dx[sec x] =

Notation. We can take derivatives of derivatives, we call those higher-order derivatives.

Second Derivative nth Derivative

f ′′(x) f (n)(x)

d2y

dx2

dny

dxn

y′′ y(n)

d2

dx2

[

f(x)] dn

dxn

[

f(x)]

D2x[y] Dn

x[y]

Higher-order derivatives have many interesting applications. One application in particular is thatwe can calculate the the curvature of the graph of a function y = f(x). Intuitively, curvature isthe amount by which a geometric object deviates from being flat and is given by

κ =|y′′|

[1 + (y′)2]3/2

Example 8. Find the curvature of y = x3.

20

3.4 Differentiating the Composition of Functions


Theorem 3.11 (The Chain Rule). If y = f(u) is a differentiable function of u and u = g(x) is a

differentiable function of x, then y = f(

g(x))

is a differentiable function of x and

dy

dx=

in other words,

d

dx

[

f(

g(x))]

=

Example 1. Findd

dx[cos(2x)]. Example 2. Find

d

dx[cot2 x].

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3 The Derivative

Example 3. Findd

dx[√

x2 + 2x + 3].

Theorem 3.12. If y =[

u(x)]n

, where u is a differentiable function of x and n is a rational number,

then

dy

dx=

in other words,

d

dx[un] =

Example 4. Findd

dx[(x3 + 27)4].

22


Example 5. Find all points on the graph of f(x) = 3

√

(x2 − 4)2 for which f ′(x) = 0 and those forwhich f ′(x) does not exist.

Example 6. Findd

dt

[

− 5

(3t + 2)2

]

.

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3 The Derivative

Example 7. Findd

dx

[

x2√

4 − x2]

.

Example 8. Findd

dx

[

x3√

x2 + 9

]

.

24


Example 9. Finddy

dxif y =

(

2x + 1

x2 + 1

)10

.

Example 10. Finddy

dxif y = sin(cos(tanx)).

25

3 The Derivative

Theorem 3.13. Let u be a differentiable function of x. Then

1.d

dx[ln x] =

2.d

dx[ln u] =

3.d

dx[ln |u|] =

Example 11. Finddy

dxif y = ln(3x). Example 12. Find

dy

dxif y = ln(x2 + 4).

Example 13. Finddy

dxif y = x2 ln x. Example 14. Find

dy

dxif y = (ln x)2.

26


Example 15. Finddy

dxif y = ln(

√x + 4).

Example 16. Finddy

dxif y = ln

[

x(x2 + 4)2

√x3 − 1

]

.

27

3 The Derivative

Recall that

• ax =

• loga x =


1.d

dx[ax] = 2.

d

dx[au] =

3.d

dx[loga x] = 4.

d

dx[loga u] =

Example 17. Finddy

dxif y = 3x. Example 18. Find

dy

dxif y = 52x.

Example 19. Finddy

dxif y = log10(sin x).

28

3.5 Implicit and Logarithmic Differentiation


We wish to differentiate functions that are defined implicitly. For example, y = −2x + 1 is saidto be in explicit form as y is already solved for as opposed to 2x + y = 1, which is in implicit

form. We have to use the Chain Rule to differentiate function to differentiate implicit functionsbecause we assume that y is a differentiable function of x. For example,

• d

dx[x2]

• d

dx[y3]

Example 1. Findd

dx[x2y3].

Example 2. Finddy

dxgiven that x3 + y3 = 6xy. Use your result to find the slope at the point (3, 3).

y

1

2

3

4

5

−1

−2

−3

−4

−5

1 2 3 4 5−1−2−3−4−5

29

3 The Derivative

Example 3. Find y′ if sin(x + y) = y2 cos x.

Example 4. Finddy

dxif y = arctan x.

30


Example 5. Finddy

dxusing logarithmic differentiation if y = xx.

Example 6. Finddy

dxusing logarithmic differentiation if y =

(

2x + 1

x2 + 1

)10

.

31

3 The Derivative

3.6 Inverse Functions and Derivatives

Theorem 3.16. Let f be a function that has an inverse, whose domain is an interval I. Then,

1. If f is continuous on I, then f−1 is continuous on its domain.

2. If f is differentiable at c and f ′(c) 6= 0, then f−1 is differentiable at f(c).

Theorem 3.17 (The Inverse Function Theorem). Let f be a function that is differentiable on

an open interval I. If f has an inverse function g, then g is differentiable at any x for which

f ′(

g(x))

6= 0 and

g′(x) =

in other words, since f−1 = g,

d

dx

[

f−1(x)]

=

Example 1. Let f(x) = 12x3 + x − 1. What is f−1(x) when x = 5? What is the value of (f−1)′(x)

when x = 5?

32

3.6 Inverse Functions and Derivatives

Note. A function and its inverse have reciprocal slopes, in other words

dy

dx=

1

dx

dy

Example 2. If f(x) = 3√

x, find the slopes of f and f−1 when x = 8.

Example 3. Suppose that f(x) = sin x and f−1(x) = arcsin x for −π/2 ≤ f(x) ≤ π/2. Use the

Inverse Function Theorem to findd

dx[f−1(x)].

33

3 The Derivative


1.d

dx[arcsin u] = 2.

d

dx[arccos u] =

3.d

dx[arctan u] = 4.

d

dx[arccot u] =

5.d

dx[arcsec u] = 6.

d

dx[arccsc u] =

Example 4. Differentiate y = arccos x − x√

1 − x2.

34

3.7 Related Rates

3.7 Related Rates

Example 1. A spherical balloon is expanding Given that the radius is increasing at at rate of 2inches per minute, at what rate is the volume increasing when the radius is 5 inches?

35

3 The Derivative

Example 2. Car A is traveling west at a rate of 50 miles per hour and car B is traveling north asa rate of 60 miles per hour/ Both are headed for the same intersection of the two roads. At whatrate are the cars approaching each other when car A is 0.3 miles and car B is 0.4 miles from theintersection respectively.

36

3.7 Related Rates

Example 3. A 13-foot ladder leans against the side of a building, forming an angle θ with theground. Given that the foot of the ladder is being pulled away from the building at at rate of 0.1feet per second, what is the rate of change of θ when then top of the ladder is 12 feet above theground?

37

3 The Derivative

Example 4. A conical paper cup of dimensions 8 inches across the top and 6 inches deep is full ofwater. The cup springs a leak and is losing water at a rate of 2 cubic inches per minute. How fastis the water level dropping when the water level is exactly 3 inches deep?

38

3.7 Related Rates

Example 5. A balloon leaves the ground 500 feet away from an observer and rises vertically at arate of 140 feet per minute. At what rate is the inclination of the observer’s line of sight increasingwhen the balloon is exactly 500 feet above the ground?

39

3 The Derivative

Example 6. A water trough with vertical cross section in the form of an equilateral triangle is beingfilled at a rate of 4 cubic feet per minute. Given that the trough is 12 feet long, how fast is the levelof the water rising when the water reaches a depth of 11

2feet?

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