CHAPTER 6

TRANSCENDENTAL FUNCTIONS

6.1 Natural Logarithm Function

• The natural logarith function, denoted by ln, is defined by

• The domaian of the natural logarithm function is the set of positive real numbers.

x

xdtt

x1

0,1

)ln(

Geometric Meaning

• Consider the graph of f(x) = 1/x. The ln(x) represents the area under f(x) between 1 and x.

Use First Fundamental Theorem to find the Derivative of the Natural

Logarithm Function

x

xx xx

xDdtt

D1

0,1

ln1

Use u-substitution within natural logarithm function

• Example:

CxCuu

du

dxxduxulet

dxx

x

)12sin(lnln2

1

2

1

)12cos(2),12sin(

)12sin(

)12cos(

Properties of natural logarithms

• If a and b are positive numbers and r is any rational number, then

• A) ln 1 = 0

• B) ln ab = ln a + ln b

• C) ln(a/b) = ln a – ln b• D) ara r lnln

Logarithmic differentiation

• Using properties of logarithms, a complicated expression can be rewritten as a sum or difference of less complicated expressions. Then, the function can be differentiated more easily.

• Example next slide.

Find g’(x)

x

xx

xxxx

xxg

xxxx

xxg

xg

xxx

x

xxxg

x

xxxg

ln

)23()1(

ln

1

23

15

1

8)('

ln

1

23

15

1

8)('

)(

1

)ln(ln)23ln(5)1ln(4

ln

)23()1(ln)](ln[

ln

)23()1()(

542

2

2

2

542

542

6.2 Inverse Functions & Their Derivatives

• If f is strictly monotonic on its domain, f has an inverse.

)()(

))((1

1

xfyyfx

xxff

Inverse Function Theorem

• Let f be differentiable and strictly monotonic on an interval I. If f’(x) does NOT euqal 0 at a certain x in I, then the inverse of f is differentiable at the corresponding point y = f(x) in the range of f and

dxdydy

dx

xfyf

1

)('

1)()'( 1

Graphical interpretation

• The slope of the tangent to a curve at point (x,y) is the reciprocal of the slope of the tangent to the curve of the inverse function at (y,x).

6.3 The Natural Exponential Function

• The inverse of ln is called the natural exponential function and is denoted by exp. Thus x = exp y, and y = ln x.

• The letter “e” denotest he unique positive real number such than ln e = 1.

The natural exponential function and the natural logarithmic function

are inverses of each other.• Properties that apply to inverse functions

apply to these 2 functions.

yallforye

xxey

x

,)ln(

0ln

The derivative of the natural exponential function is itself

xx

xxx

edxe

eeD )(

Integrate the following:

Ce

Cuu

du

dxedueulet

dxe

e

x

xx

x

x

223

3

)2(2

1

2

1

,2

)2(

6.4 General Exponential & Logarithmic Functions

1,ln

1

ln

ln)ln()ln( ln

ln

aCaa

dxa

aaaD

axea

ea

xx

xxx

axx

axx

Logarithms could have a base other than e.

1,1

ln

1log

ln

lnlog

1

aCa

xdxx

axxD

a

xx

aa

ax

a

Example:

CCdu

xdxduxulet

xdx

xuu

x

5ln

55

5ln

15

sec,tan

sec5

tan

2

2tan

6.5 Exponential Growth & Decay

• Functions modeling exponential growth (or decay) are of this form:

0:,0:

kdecaykgrowth

eyy kto

Compound InterestA = amount

r = interest raten = # times compounded

t=time in years

rto

nt

o

eAtAlycontinuousCompound

n

rAtA

)(:

1)(

6.6 1st-Order Linear Differential Equations

• Sometimes it is not possible to separate an equation such that all expressions with x and dx are on one side and y and dy are on the other.

• General form of a first-order linear differential equation:

operatoridentityI

operatorderivativeD

xQIxPyD

x

yx

)()(

Example: Solve the differential equation:xy’(x) – 2y(x) = 2

1)(

)(

2)]([

2)(2)('

)(

2)(

2)('

2

22

32

332

2lnln22

2

Cxxy

Cxxyx

xxyxdx

d

xxyxxyx

xeeebythroughmultiply

xxy

xxy

xxdx

x

6.7 Approximations for Differential Equations

Slope fields: Consider a first-order differential equation of the form y’ = f(x,y)

At the point (x,y) the slope of a solution is given by f(x,y).

Example:y’ = 3xy, at (2,4), y’=24, at (-2,1), y’=-6; at (0,5), y’ = 0; at (2,0), y’=0, etc.

If all the slopes (y’) were graphed on a coordinate axes at those specific points, the resulting graph would be a “slope field”.

Approximating solutions of a differential equation

• Euler’s Method: To approximate the folution of y’ = f(x,y) with initial condition y(x-not)=y-not, choose a step size ha nd repeat the following steps for n = 1,2,3,…

),(.2

.1

111

1

nnnn

nn

yxhfyySet

hxxSet

Applying Euler’s Method

• Use your calculator and the table function to evaluate the function until the solution is found with the desired error.

6.8 Inverse Trigonometric Functions & Their Derivatives

• If the domain of the trigonometric functions is restricted, a portion of the curve is monotonic and has an inverse.

xxyyx

xxyyx

xxyyx

xxyyx

0,secsec

22,tantan

0,coscos

22,sinsin

1

1

1

1

Using triangles, some useful identities are established.

1,1

1,1)tan(sec

1)sec(tan

1)cos(sin

1)sin(cos

2

21

21

21

21

xx

xxx

xx

xx

xx

Derivatives of 4 Inverse Trigonometric Functions

1,1

1sec

1

1tan

11,1

1cos

11,1

1sin

2

1

21

2

1

2

1

xxx

xD

xxD

xx

xD

xx

xD

x

x

x

x

Corresponding integral formulas follow from these derivatives

Cxdxxx

Cxdxx

Cxdxx

1

2

12

1

2

sec1

1

tan1

1

sin1

1

Example

CxCx

CxCuduu

xdxduxulet

dxx

x

)5(5

1

)5(tantan5

1tan5

1

1

1

5

1

sec5),5tan(

)5(tan1

)5(sec

112

2

2

2

6.9 Hyperbolic Functions & Their Inverses

)sinh(

1)(csc,

)cosh(

1)(sec

)sinh(

)cosh()coth(,

)cosh(

)sinh()tanh(

2)cosh(,

2)sinh(

xxh

xxh

x

xx

x

xx

eex

eex

xxxx

Derivatives of Hyperbolic Functions

)coth()(csc)(csc

)tanh()(sec)(sec

)(csc)coth(

)(sec)tanh(

)sinh()cosh(

)cosh()sinh(

2

2

xxhxhD

xxhxhD

xhxD

xhxD

xxD

xxD

x

x

x

x

x

x

Example

CxCuu

du

xdxduxulet

dxx

xdxx

sinhlnln

cosh,sinhsinh

cosh)coth(