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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