2.8 Implicit Differentiation

Definition. We will say that a given equation in x and y defines the function f implicitly if the graph of y = f(x) coincides with a portion of the graph of the equation.

Example:

•The equation implicitly defines functions

2 2 1x y 2 2

1 2( ) 1 and ( ) 1f x x f x x

2x y

1 2( ) and ( )f x x f x x

•The equation implicitly defines the functions

There are two methods to differentiate the functions defined implicitly by the equation.

For example: Find / if 1dy dx xy

One way is to rewrite this equation as , from which it

follows that

1y

x

2

1dy

dx x

Two differentiable methods

With this approach we obtain [ ] [1]

[ ] [ ] 0

0

d dxy

dx dxd d

x y y xdx dxdyx ydxdy y

dx x

The other method is to differentiate both sides of the equation before solving for y in terms of x, treating y as a differentiable function of x. The method is called implicit differentiation.

Since , 1

yx

2

1dy

dx x

Two differentiable methods

Implicit Differentiation

Example: Use implicit differentiation to find dy / dx if

Solution: 2 2

2 2

[ ] [3 ]

[ ] [ ] 3

2 2 3

2 3 2

3 2

2

d dx y x

dx dxd dx y

dx dxdy

x ydx

dyy xdx

dy x

dx y

2 2 3x y x

Example

Example: Find dy / dx if 3 3 11 0y x

Solution: 3

3

2

2

2

[ 3 11] [0]

[ ] 3 [ ] [11] [0]

2 3 0

2 3

3

2

d dy x

dx dxd d d dy x

dx dx dx dxdy

ydxdy

ydx

dy

dx y

Example

2.10 Logarithmic Functions

Logarithm Function with Base a

Natural Logarithm Function

Logarithms with base e and base 10 are so important in applications thatCalculators have special keys for them. logex is written as lnx log10x is written as logx

The function y=lnx is called the natural logarithm function, and y=logx isOften called the common logarithm function.

Properties of Logarithms

Properties of ax and logax

Derivative of the Natural Logarithm Function

3Find [ln( 4)]d

xdx

3Let 4, we obtainu x

Example:

Solution:

3 33

23

2

3

1[ln( 4)] [ 4]

41

= (3 )4

3 =

4

d dx x

dx x dx

xx

x

x

Note: 1

[ln ] , 0d

x xdx x

Example

Example:

1 1[ln | cos |] [cos ] ( sin ) tan

cos cos

d dx x x x

dx x dx x

Find [ln | cos |]d

xdx

Solution:

Derivatives of au

Example: 2 2 22[ ] [ ] 2x x xd de e x xe

dx dx

sin sin sin[3 ] 3 ln3 [sin ] ln3cos 3x x xd dx x

dx dx

Note that [ ] ln , [ ] , [ ]x x x x u ud d d dua a a e e e e

dx dx dx dx

Derivatives of logau

2

1 3 1[log (3 )] [3 ]

3 ln 2 3 ln 2 ln 2

d dx x

dx x dx x x Example:

10

1 1[log ( )] [ ]

ln10 ln10 ln10

xx x

x x

d d ee e

dx e dx e

Note that 1

lnd du

udx u dx

The Number e as a Limit

2.11 Inverse Trigonometric Functions

The six basic trigonometric functions are not one-to-one (their values Repeat periodically). However, we can restrict their domains to intervals on which they are one-to-one.

Six Inverse Trigonometric Functions

Since the restricted functions are now one-to-one, they have inverse, which we denoted by

These equations are read “y equals the arcsine of x” or y equals arcsin x” and so on.

Caution: The -1 in the expressions for the inverse means “inverse.” It doesNot mean reciprocal. The reciprocal of sinx is (sinx)-1=1/sinx=cscx.

1

1

1

1

1

1

sin arcsin

cos arccos

tan arctan

cot arccot

sec arcsec

csc arccsc

y x or y x

y x or y x

y x or y x

y x or y x

y x or y x

y x or y x

Derivative of y = sin-1x

Example: Find dy/dx if

Solution:

1 2sin ( )y x

2 2 2 2

1 2(2 )

1 ( ) 1 ( )

dy xx

dx x x

Derivative of y = tan-1x

Example: Find dy/dx if

Solution:

1tan ( )xy e

2 2

1( )

1 ( ) 1 ( )

xx

x x

dy ee

dx e e

Derivative of y = sec-1x

Example: Find dy/dx if

Solution:

1 3sec (4 )y x

1 4 4

4 4 2

3 3

4 8 4 8

8

1[sec (3 )] (3 )

| 3 | (3 ) 1

12 12

| 3 | 9 1 3 9 1

4

9 1

dy d dx x

dx dx dxx x

x x

x x x x

x x

Derivative of the other Three

There is a much easier way to find the other three inverse trigonometric Functions-arccosine, arccotantent, and arccosecant, due to the followingIdentities:

It follows easily that the derivatives of the inverse cofunctions are the negativesof the derivatives of the corresponding inverse functions.