MTH 251 – Differential Calculus

Chapter 3 – Differentiation

Section 3.8

Derivatives ofInverse Functions and Logarithms

Copyright © 2010 by Ron Wallace, all rights reserved.

Review: Inverse Functions

• If y = f(x) is a one-to-one function, then there is

a function y = f -1(x) … called the inverse function … such that …

• That is, the inverse function “un-does” the function and vice-versa.

• Examples ………..

1 1( ( )) ( ( ))f f x x f f x

2( ) , 0f x x x

1( )f x x

( ) xf x e

1( ) lnf x x

1( ) xf x

1 1( ) xf x

Finding the Inverse of a Function

• In many cases, the inverse can be found by …1. Writing the function as y = f(x)

2. Switch the variables … i.e. x = f(y)

3. Solve for y.

• Examples … find the inverses of …

2 5y x

1

1

xy

x

Finding the Inverse of a Function

• In other cases, the inverse cannot be found algebraically and therefore the existence of such an inverse is recognized and given a name.

• Examples …

( ) , 0 & 1xf x a a a

2 2( ) sin , f x x x

Derivatives of Inverse Functions

• What is the relationship between the derivative of a function and the derivative of its inverse?

• Begin with two simple examples …

( ) 2 5f x x 1 512 2( )f x x

2 5 2d

xdx

51 12 2 2

dx

dx

2( ) 5, 0f x x x 1( ) 5f x x

2 5 2d

x xdx

1

2 55

x

dx

dx

Reciprocal?

Reciprocal?Almost!

Derivatives of Inverse Functions

• The derivative of the inverse of a function is the reciprocal of the derivative of the original function evaluated at the inverse.

• Or … since the inverse of the inverse is the original function …

1

1

( )

1( )

( )f x

df x

ddx f xdx

1

( )

1( )

( )f x

df x

ddx f xdx

This may be the more

practical form.

Derivatives of Inverse Functions

• Proof

1

1

( )

1( )

( )f x

df x

ddx f xdx

1 1' ( ) ( ) 1d

f f x f xdx

1

1

1

( )

1 1( )

' ( ) ( )f x

df x

ddx f f x f xdx

1( )f f x x 1( )d d

f f x xdx dx

Derivatives of Inverse Functions

• Example …

1

( )

1( )

( )f x

df x

ddx f xdx

dx

dx

NOTE: Where we really need this is when we cannot algebraically find the inverse of a function (i.e. exp, log, trig, & others).

Derivative of the Natural Log

• Since the inverse of the natural log is the exponential function …

1

( )

1( )

( )f x

df x

ddx f xdx

lnd

xdx

ln

1

x

x

de

dx

ln

1x

xe

ln

1xe

1

x

Memorize this result1

: l.N T O nEd

xdx x

Derivative of the Natural Log Second form

lnd

xdx

This gives the more general

1ln

re

,

s .

t

ul

0d

x xdx x

? Note: x < 0

1 11

x x

Using the chain rule.

( ) lnf x x

The Chain Rule and theDerivative of the Natural Log

• Examples …

'( )ln( ( ))

( )

d f xf x

dx f x

ln5d

xdx

ln(sin )d

xdx

3 2ln(2 4 1)d

x x xdx

Memorize this one!

Derivatives of other Logs

loga

dx

dxln

ln

d x

dx a

Change of base formula.

1ln

ln

dx

a dx ln a is just a

constant

1

lnx a

Yes … you need to memorize this one too!

Review: Laws of Logarithms

log ( ) log ( ) log ( )a a amn m n

log ( ) log ( ) log ( )ma a an m n

log ( ) log ( )na am n m

Logarithmic Differentiation

• Due to the laws of logarithms, it is often easier to find the derivative of the log of a function instead of the derivative of the original function.

• Steps … find the derivative of y = f(x)1. Take a logarithm of both sides (use natural logs)

2. Expand ln(f(x)) using the laws of logarithms

3. Differentiate implicitly

4. Solve for dy/dx• This will simply require multiplying both sides of the

equation by y or f(x).

Logarithmic Differentiation

• Example 1 of 3 …

3(2 1)

sin

d x

dx x x

Logarithmic Differentiation

• Example 2 of 3 … (a more practical application)

xdx

dx

Logarithmic Differentiation

• Example 3 of 3 … (a more practical application)

sin xdx

dx

The Power Rule … One Last Time!

• What if n is a real number?

1, 0 & n ndx nx x n Q

dx

lnn n xx e

lnn n xd dx e

dx dx ln lnn x d

e n xdx

lnn x ne

x n n

xx

1nnx

0 & x n R