Section 4.4The Chain Rule

F' x f ' xg x g'

F x xf g

F' x f ' xg x g'

4xx xF ln

3 11x lnx 4

xxF'

F x xf g

Find f ‘ (x) if 32f x x 4x

2 22x3 xf ' 4x 4x

Find f ‘ (x) if 2

2x 3f x

5x 2

2

2 5x 2 5 2x 3

5x

2x 3

5x 22

2f ' x

3

2x 3f ' x 22

5x 2

Try these two…

NO CALCULATOR

3 / 42If h x x 4 1, then h' 2

A) 3 B) 2 C)1 D) 0 E) DNE

2 1/42x 4

3h' x

4x

1/43h' 02 4

4

NO CALCULATOR

2 2If y cos x sin x, then y '

A) 1 B) 0 C) 2 cos x sinx D) 2 cos x sinx E) 4cos x sinx

y cos2x

y ' sin2x 2

y ' 4sinxcos x

5

kx5

5 k 5 kx kx x kx

d yIf y e , then

dx

A) k e B) k e C) 5!e D) 5!e E) 5e

kx kx 2 kxy e y ' ke y" k e etc.

NO CALCULATOR

What is the instantaneous rate of change at x = 0 of the functionf given by 2xf x e 3sinx A) 2 B) 1 C) 0 D) 4 E) 5

2xf ' x 2e 3cos x

02f ' 2e0 s 03co

The y-intercept of the tangent line to the curve y x 3 at 1, 2 is 1 1 3 5 7

A) B) C) D) E)4 2 4 4 4

1/ 2dy 1x 3

dx 2

x 1

dy 1|

dx 4

1y 2 x 1

4

10y 2 1

4

2 x

x x 2 x

x 2 x

2 x x

x 2 x

x x

If g x tan e , g' x

A) 2e tan e sec e

B) 2tan e sec e

C) 2tan e sec e

D) e sec e

E) 2e tan e

NO CALCULATOR

xg' x 2tan e 2 xsec e xe

NO CALCULATOR

2

2

2

2 2

If h x f x f x g x , , and , then h' x

A) f x g x

B) 2f x f x g x

C) f x g x

D) f x g x

g' x f x

E) g x 2

f ' x g

f x g x f x

x

2h x f x f x g x

h' x 2 f x f ' x f ' x g x g' x f x

g xh' f xx 2 f x g x f xg x

CALCULATOR REQUIRED

Let the function f be differentiable on the interval [0, 2.5] and Use the table to estimate g ‘ (1) if g x f f x

x 0 0.5 1 1.5 2 2.5f(x) 1.7 1.8 2 2.4 3.1 4.4

A) 0.8 B)1.2 C)1.6 D) 2.0 E) 2.4

g x f f x

g' x f ' f x f ' x 1 1 f 1g' f ' f ' g' f '1 12 f '

g' 1 2 0.6

3x

3x 3x

dlne

dx1 3

A)1 B) 3 C) 3x D) E)e e

y 3xlne

NO CALCULATOR

2t

The formula x t ln t 1 gives the position of an object 18

moving along the x-axis during the time interval 1 t 5. At

the instant when the acceleration of the object is zero, the

velocity is

1 2A) 0 B) C)

3 3

D)1 E) undefined

NO CALCULATOR

2t

x t ln t 118

1 1v t t

t 9 2

1 1a t

t 9

2

1 10

t 9

t 3 t 3

1 1v 3 3

3 9

2

2

2 2

The slope of the line tangent to the graph of y ln x at e ,1 is

e 2 1 1 1A) B) C) D) E)

2 e 2e 2e e

dy 1

dx 2x

2 2e ,1

dy 1|

dx 2e

NO CALCULATOR

1y lnx

2

If f x ln cos2x , then f ' x

A) 2tan2x B) cot 2x C) tan2x D) 2cot 2x E) 2tan2x

1f ' x

cos2x sin2x 2

2sin2xf ' x

cos2x

NO CALCULATOR

NO CALCULATOR

If f x sin2x ln x 1 , then f ' 0

A) 1 B) 0 C)1 D) 2 E) 3

1f ' x cos2x 2

x 1

1f ' 0 cos 0 2

0 1

NO CALCULATOR

g x

2x 1

If e 2x 1, then g' x

1 2A) B) C) 2 2x 1 D) e E) ln 2x 1

2x 1 2x 1

g xln le 1n 2x

lng x 2x 1

1g' x 2

2x 1

NO CALCULATOR

2

1/ 2 3 / 2 2

A particle moves on the x-axis in such a way that its position

at time t, t 0, is given by x t lnx . At what value of t does

the velocity of the particle attain its maximum?

A)1 B) e C) e D) e E) e

2x t lnx 1 2lnx

v t 2lnxx x

2

2x 1 2lnx

xa t

x

2

2 2lnx0

x

lnx 1

2

2

2 2 2

d yIf y ln cos x and 0 x , what is in terms of x?

2 dx

A) tanx B) tanx C) sec x D) sec x E) csc x

NO CALCULATOR

dy 1sinx tanx

dx cos x

2

22

d ysec x

dx

2 2xIf f x ln x e , then f ' 1

A) 0 B)1 C) 2 D) e E) undefined

NO CALCULATOR

2x2 2x

1f ' x 2x 2e

x e

2

2

2 2ef ' 1

1 e

2x

2 2

If f x e and g x lnx, then the derivative of y f g x at x e is

A) e B) 2e C) 2e D) 2 E) undefined

y f y ' f ' g' xg x g x

2x 1g'f ' x 2e x

x

2lnxf ' 2eg x

2lne2e

f g' ee

g e'

NO CALCULATOR

CALCULATOR REQUIRED

The position of a particle moving on the x-axis, starting at t = 0, is given by 3

x t t a t b where 0 a b

Which of the following statements is true?I. The particle is at a positive position on the x-axis at time t = (a + b)/2II. The particle is at rest at time t = aIII. The particle is moving to the right at time t = b.

A) I only B) II only C) III only D) I and II only E) II and III only

3x t t 1 t 2

33 3 3

x 1 22 2 2

NO

2 3x ' t 3 t 1 t 2 t 1

2 3x ' 1 3 1 1 1 2 1 1

YES

2 3x ' 2 3 2 1 2 2 2 1

YES

f xLet h x f g x and k x

g x . Fill in the chart below.

x f(x) f ' (x) g(x) g ' (x) h(x) h ' (x) k(x) k ' (x)-1 -1 4 1 -1 8 -10 1 0 0 0 2 01 -4 1 -1 -8 -1

h 1 f g 1 1 f 1

1

h 0 f g 0

h 0 f 0

h 0 1

h' x f ' g x g' x h' 1 f ' g 1 g' 1

8 f ' 1 g' 1

8 4g' 1

-2

h' 1 f ' g 1 g' 1 8 f ' 1 g' 1

8 4g' 1

21

h' 0 f ' g 0 g' 0 h' 0 f ' 0 g' 0

h' 0 0 0

0

P. 225 #45 Finish