Deck 2: The Derivative

Question 1

Find f '(x) where .

Answer 1

Question 2

If , (Do not reduce any trig ratio identities.)

A)
B)
C)
D)
E)

Answer 2

Question 3

Find an equation for the tangent line to the graph of y = x sin x at x = $\pi$

A)
B)
C)
D)
E)

Answer 3

Question 4

Find f'(x) where f(x) = x²(sin 8x)⁶.

Question 5

Answer true or false. If , .

Question 6

y = cot (sin x). Find dy/dx.

A) cos x csc² (sin x)
B) - cos x csc² (sin x)
C) - sin x csc² (cos x)
D) sin x csc² (cos x)
E) - sin x csc² (sin x)

Question 7

If , then

A)
B)
C)
D)
E)

Question 8

Find the value of the constant A in y = A sin 3t so that d²y/dt² + 7y = sin 3t.

A)
B)
C)
D) 2
E) 16

Question 9

Answer true or false. If y = cos(x⁶), d²y/dx² = -cos(x⁶).

Question 10

Use a graphing utility to obtain the graph of . Determine the slope of the tangent line to the graph at x = 0.

A)
B) 0
C) 3
D) 16
E) 8

Question 11

f(x) = (x⁴ - 3)³⁸. f '(x) =

A) 38x³ (x⁴ - 3)³⁷
B) 4x³ (x⁴ - 3)³⁷
C) 152x³ (x⁴ - 3)³⁷
D) 152(x⁴ - 3)³⁷
E) 38(x⁴ - 3)³⁷

Question 12

Answer true or false. Given f '(x) = 5x and , then if F(x) = f(g(x)).

Question 13

y = sin³( $\pi$ - 9 $\theta$ ). Find dy/d $\theta$ .

A) - 27 cos( $\pi$ - 9 $\theta$ ) sin²( $\pi$ - 9 $\theta$ )
B) - 27 cos( $\pi$ + 9 $\theta$ ) sin²( $\pi$ + 9 $\theta$ )
C) - 27 cos ²( $\pi$ - 9 $\theta$ ) sin( $\pi$ - 9 $\theta$ )
D) - cos( $\pi$ - 9 $\theta$ ) sin²( $\pi$ - 9 $\theta$ )
E) - 27 sin²( $\pi$ - 9 $\theta$ )

Question 14

Find dy/dx if y = cos(sin x).

A) -cos x sin(cos x)
B) -sin(sin x)
C) cos x sin(sin x)
D) -cos x sin(sin x)
E) sin(cos x)

Question 15

Answer true or false. If y = sin 6x - cos x³, d²y/dx² = 18 cos x.

Question 16

y = x⁴ sin(5x). Find dy/dx.

A) 5x⁴ cos(5x) - 4x³ sin(5x)
B) 5x⁴ cos(5x) + x³ sin(5x)
C) 5x⁴ cos(5x) + 4x³ sin(5x)
D) 5x⁴ cos(5x)
E) 4x³ sin(5x)

Question 17

Find f '(x) where f(x) = cos(tan 4x).

Question 18

Find f '(x) if f(x) = (x⁵ - 11)¹⁰.

A) 10x⁴(x⁵ - 11)⁹
B) 5x⁴(x⁵ - 11)⁹
C) 50(x⁵ - 11)⁹
D) 50x⁴(x⁵ - 11)⁹
E) 10(x⁵ - 11)⁹

Question 19

If ,

A) cos x
B) - sin x
C) - cos x
D) sin x
E) sin x cos x

Question 20

f(x) = sin(6x). f'(x) =

A) 6 cos(6x)
B) cos(6x)
C) 6 cos(x)
D) 6 sin(6x)
E) sin(6x)

Question 21

Find f '( $\theta$ ) where f( $\theta$ ) = ( $\theta$ + sin 2 $\theta$ )².

Question 22

Find f '(x) where f(x) = (x⁵ - 3)⁷(x⁵ + 2)⁴.

Question 23

Find equations for the tangent and normal lines to the graph of f(x) = cos(4 - x²) at x = 2.

Question 24

Find f '( $\theta$ ) where f( $\theta$ ) = cos²(6 $\theta$ ⁶ - $\theta$ )⁶.

Question 25

Find y '( $\pi$ ) where .

Question 26

Find f '(x) if .

Question 27

Find where y = (6x + 3)⁶(2x + 3)⁴.

Question 28

Find where f(x) = sin ⁶ 5x.

Question 29

Find f '(x) where .

Question 30

Find where

Question 31

Find equations for the tangent and normal lines to the graph of f(x) = x cos 6x at x = $\pi$ /6.

Question 32

Given that f(1) = 5, f '(1) = 4 and g(x) = (f(x))^-4, find g '(1).

Question 33

y = x³ cos(3x). Find dy/dx.

Question 34

Find f '(t) where .

Question 35

Find f '(x) where f(x) = (x⁶ + 6x)¹¹.

Question 36

Find f '(x) where .

Question 37

Find f '(x) where .

Question 38

Find (f $\circ$ g)'(3) if f '(3) = 5, g(3) = 3 and g '(3) = 2.

Question 39

Find equations for the tangent and normal lines to the graph of f(x) = sin(16 - x²) at x = 4.

Question 40

Find f '(x) if f(x) = cos(x⁶).

Question 41

Answer true or false. If y = sin x, d²y/dx² = sin x.

Question 42

Find the equation of the line tangent to the graph of y = sin x at the point where x = 0.

A) y = cos x
B) y = x
C) y = -x

Question 43

Answer true or false. If , .

Question 44

If y = x⁷ cos x, find dy/dx.

A) 7x⁶ cos x + x⁷ sin x
B) 7x⁶ sin x - x⁷ cos x
C) 42x⁵ cos x - 14x⁶ sin x - x⁷ cos x
D) 42x⁵ cos x - 14x⁶ sin x
E) 7x⁶ cos x - x⁷ sin x

Question 45

Answer true or false. If x is given in radians, the derivative formula for y = sin x in degrees is .

Question 46

Find d⁶y/dx⁶ if y = cos x.

A) -cos x
B) cos x
C) sin x
D) -sin x
E) sin x cos x

Question 47

Find f '(x) if f(x) = cos x tan x.

A) sec x
B) - cos x
C) - sin x tan x + 1
D) cos x
E) - tan x

Question 48

Answer true or false. is differentiable everywhere.

Question 49

Find all x-values on [0, 2 $\pi$ ] where f(x) is not differentiable. f(x) = cos x cot x

A)
B) 0, $\pi$ , 2 $\pi$
C)
D) 0
E) none

Question 50

Find d²y/dx² if y = x tan x.

A) 2x tan x sec x
B) x sec x
C) 2 sec ² x + 2x tan x sec ² x
D) sec x
E) -2x tan x sec x

Question 51

Find d ⁷⁵ of cos x/dx⁷⁵.

A) -cos x
B) -sin x
C) cos x
D) 0
E) sin x

Question 52

Find f '(x) if f(x) = x⁴ cos x.

A) 4x³ cos x - x⁴ sin x
B) 4x³ cos x
C) 4x³ sin x
D) -4x³ cos x
E) -4x³ sin x

Question 53

If y = x⁶ cos x, find d²y/dx².

A) 30x⁴ cos x - 12x⁵ sin x + x⁶ cos x
B) 30x⁴ cos x - 6x⁵ sin x - x⁶ cos x
C) 30x⁴ cos x - 12x⁵ sin x - x⁶ cos x
D) 30x⁴ cos x - 12x⁵ sin x
E) 6x⁵ cos x - x⁶ sin x

Question 54

Find the x-coordinates of all points in the interval [-2 $\pi$ , 2 $\pi$ ] at which the graph of has a horizontal tangent line.

A)
B)
C)
D)
E)

Question 55

Find f '(x) if .

Question 56

Find f '(x) if f(x) = x tan x.

Question 57

A rock at an elevation angle of $\theta$ is falling in a straight line. If at a given instant it has an angle of elevation of $\theta$ = and is a horizontal distance s from an observer, find the rate at which the rock is falling with respect to $\theta$ .

A)
B)
C)
D)
E)

Question 58

Find f '(x) if f(x) = x sin x.

Question 59

Answer true or false. If f(x) = cos x sec x - sin x, f '(x) = 1 - cos x.

Question 60

Find f '(x) if f(x) = csc²x + cot²x.

A) 2 csc x + 2 cot x
B) 2 csc x - 2 cot x
C) -1
D) 1
E) 0

Question 61

Differentiate .

Question 62

Find if .

Question 63

Find y''(x) if .

Question 64

Show by use of a trigonometric identity that .

Question 65

A 16 foot long ladder leans against a wall at an angle $\theta$ with the horizontal as shown in the figure. The top of the ladder is x feet above the ground. If the bottom of the ladder is pushed toward the wall, find the rate at which x changes with $\theta$ when $\theta$ = 45°. Express the answer in units of feet/degree.

Question 66

f(x) = 3x cot x. Find f '(x).

Question 67

An airplane is flying on a horizontal path at a height of 6000ft, as shown in the figure. At what rate is the distance s between the airplane and the fixed point P changing with $\theta$ when $\theta$ = 20°. Express the answer in units of feet/degree.

Question 68

Find if .

Question 69

Find if .

Question 70

Find f '(x) if f(x) = (7x⁴ + 9) cos x.

Question 71

Find f '( $\theta$ ) if .

Question 72

Answer true or false. If , .

Question 73

Find f'( $\theta$ ) if .

Question 74

f(x) = 4 cos x + 7 sin x. Find f'(x).

Question 75

Find if y = tan x .

Question 76

f(x) = x² cos x + sin x. Find f '(x).

Question 77

Find if y = (3x² + 5x) cot x.

Question 78

Find if .

Question 79

Find f '(x) if .

Question 80

Find if y = 8x sin x - 3 cos x + x² cos x.