QuestionIf f is differentiable, find the limit

Sol

.)()3(

lim22

0 h

hxfhxfh

).()(8)]()([)(4

)]}()3([)()3(

{lim

)()3(lim

0

22

0

xfxfxfxfxf

hxfhxfh

hxfhxfh

hxfhxf

h

h

Question Find the limit:

(1) (2)

Sol. (1)

(2)

0

sin 4lim

sin 6x

x

x 0

tan 6lim

sin 2x

x

x

0 0

sin 4 sin 4 6 4 2 2lim lim 1 1

sin 6 4 sin 6 6 3 3x x

x x x

x x x

0 0

tan 6 sin 6 2 1 6lim lim 1 1 1 3 3

sin 2 6 sin 2 cos 6 2x x

x x x

x x x x

The Chain RuleTheorem If u=g(x) is differentiable at x=a and y=f(u) is

differentiable at u=g(a). Then y=f(g(x)) is differentiable at

x=a, and

or,

[ ( ( ))] ( ) ( ) ( ( )) ( ),f g x f u g x f g x g x

dy dy du

dx du dx

Derivative of power functionsEx. Differentiate

Sol. Since can be written as the composition

by the chain rule, we have

.y x

ln xy e

ln 1( ) ( ) u a xy f u g x e e x xx x x

( ) , ( ) ln ,uf u e u g x x

Derivative of exponential functions

Ex. Differentiate

Sol. Since can be written into

by the chain rule, we have

xy a

lnx x ay a e

ln( ) ( ) ln ln lnu x a xy f u g x e a e a a a

( ) , ( ) (ln ) ,uf u e u g x a x

ExampleEx. Differentiate

Sol. Let then

By the chain rule, we have

2( ) 1F x x

2( ) , ( ) 1,f u u u g x x ( ) ( ( )).F x f g x

1/ 2

2

1( ) ( ) ( ) 2

2 1

x xF x f u g x u x

u x

The power rule The power rule combined with the chain rule

Ex. Find the derivative of

Sol.

1[ ( )] ( ) ( )n nf x nf x f x 9

2( )

2 1

tg t

t

8

2 2( ) 9

2 1 2 1

t tg t

t t

8 8

2 10

2 (2 1) 1 ( 2) 2 45( 2)9

2 1 (2 1) (2 1)

t t t t

t t t

The chain ruleIf y=f(u), u=g(v) and v=h(x) are all differentiable, then

y=f(g(h(x))) is differentiable and

or,

[ ( ( ( )))] ( ) ( ) ( ) ( ( ( ))) ( ( )) ( )f g h x f u g v h x f g h x g h x h x

dy dy du dv

dx du dv dx

Example Ex. Differentiate

Sol.

( ) sin(cos(tan ))f x x

2( ) cos(cos(tan )) [ sin(tan )] secf x x x x

2cos(cos(tan )) sin(tan ) secx x x

Logarithmic differentiationEx. Find the derivative of

Sol. Not a power function, not an exponential function

Since by product rule and chain rule,

The method used here is called logarithmic differentiation

2tan ln(1 ) 2 22

2 tan 2 22

2[sec ln(1 ) tan ]

12 tan

(1 ) [sec ln(1 ) ].1

x x

x

xy e x x x

xx x

x x xx

2 tan(1 ) .xy x

2tan ln(1 ) ,x xy e

Logarithmic differentiationIn general, to differentiate we can take logarithm

first to get then differentiating both sides

Question: Find the derivative of

Sol.

( ) .1

xx

f xx

( )( ) ,g xy f xln ( ) ln ( ),y g x f x

( )( ) ( )( ) ln ( ) ( ) ( ) [ ( ) ln ( ) ( ) ].

( ) ( )g xy f x f x

g x f x g x y f x g x f x g xy f x f x

1( ) ln .

1 1 1

xx x

f xx x x

Question Differentiate

Sol.

2 3

23

( 1) 3 2.

( 3)

x xy

x

2

5 23

( 1)(15 74 31).

3 ( 3) (3 2)

x x xy

x x

Implicit differentiation Materials in textbook: page 227-233

Outline Derivative of implicit functions Derivative of inverse trigonometric functions

Expressions of functions Explicit expression: y can be explicitly expressed in term

of x, for example,

Implicit expression: x and y related by an equation, and can not solve y in terms of x explicitly, for example,

3 1y x

4 3 arcsin .xyx y e x

Implicit differentiationEx. Find if

Sol. Differentiating both sides with respect to x, regarding y

as a function of x, and using the chain rule, we get

Solving the equation for we obtain

Ex. Find an equation of the tangent line to the curve

at the origin.

Sol. is the slope

y 3 3 6 .x y xy

2 23 3 6 6 .x y y y xy ,y

2

2

2.

2

y xy

y x

4 3 2 3 x y x y4 2 3 3 2

3 4 2 3 (0)3

x y y x y y y

Example Suppose y=f(x) is defined implicitly by

(1) Find (2) Let find

Sol. (1)

(2)

ln 1. xy y( )( ) (ln ) , f xg x f x e( )f x (1).g

21 ( )( ) ( ) ( ) 0 ( )

( ) 1 ( )

f xf x xf x f x f x

f x xf x

( ) ( )

(1) (1)

1( ) (ln ) (ln ) ( )

(1) (0) (0) (1).

f x f x

f f

g x e f x f x e f xx

g e f f e f2 1

(0) , (1) 1, (0) , (1) .2

f e f f e f2 31

(1) .2

g e e

Homework 5 Section 3.1: 45, 56, 57

Section 3.2: 10, 21, 42

Section 3.4: 11, 16, 38, 39, 42

Section 3.5: 20, 28, 40

Section 3.6: 10, 18

Derivative of arcsine function Ex. Find the derivative of

Analysis. means

differentiating will give

Sol. Differentiating implicitly with respect to x,

we obtain so

sinx yarcsiny x( )y x

arcsin .y x

2

1(arcsin )

1x

x

sinx ysinx y

1 cos y y

2

1 1

cos 1y

y x

Derivative of inverse functions If x=f(y) is differentiable and then the inverse

function is differentiable and

or,

Proof.

1 1[ ( )] ,

( )f x

f y

( ) 0,f y 1( )y f x

1.

dydxdxdy

1( ) 1 ( ) ( ) ( )

( )x f y f y y x y x

f y

Example Similarly,

22

2

1 1(arccos ) , (arctan ) ,

111

( cot ) .1

x xxx

arc xx

Higher derivatives The derivative of is called the second derivative of

f and denoted by or

Recursively, we can define the third derivative and generally the nth derivative

Interpretation: for example, if s(t) is displacement, then is velocity, is acceleration and is jerk.

( )f x( ) , f f

22

2( ) , ( )

d dy d yD f x

dx dx dx

( ) , f f

( 1) ( )( ) ( ). n

n n nn

d yf f D f x

dx( )s t

( )s t( )s t

ExampleIf

then

3 26 5 3y x x x

23 12 5y x x

6 12y x

6y( ) 0 ( 4)ny n

ExampleFind if

Sol.

At x=0, y=1, and

thus

. ye xy e(0)y

2

0

( ) (1 )

( )

yy

y y

y

ye y y xy y

x e

y x e y e yy

x e

0

1(0) ,

y

x

yy

x e e

2 2

0

( ) (1 ) 1(0) .

( )

y y

y

x

y x e y e yy

x e e

ExampleIf find

Sol.

( ) ( )nf x1

( ) ,f xx

11( )f x x

x 2( )f x x

3 3( ) ( 1)( 2) 2f x x x

4 4( ) ( 1)( 2)( 3) 3!f x x x

( ) 1 1( ) ( 1)( 2)( 3) ( ) ( 1) !n n n nf x n x n x

Example Find if

Sol. Using the trigonometry identity

Suppose then

Therefore

( )ny

cos sin( ),2

y x x

sin .y x

cos sin( ),2

cos( ) sin( 2 ).2 2

y x x

( ) sin( ),2

ky x k

( 1) cos( ) sin( ( 1) ).2 2

ky x k x k

( )(sin ) sin( ).2

n nx x

Question Find if

Hint:

Sol.

( )ny 2

1.

2

y

x x

2

1 1 1 1( ).

2 3 1 2

x x x x

( )1 1

( 1) ! 1 1[ ].

3 ( 1) ( 2)

nn

n n

ny

x x