2.3: Product and Quotient Rules, Higher-Order Derivatives

Question 1

d/dx[f(x)⋅g(x)] = ?

Answer 1

f’(x) g(x) + f(x) g’(x) = f’g + fg’

Question 2

d/dx[f(x) / g(x)] = ?

Answer 2

f’(x) g(x) - f(x) g’(x) / (g(x))^2 = f’g - fg’ / g^2

Question 3

d/dx[sin(x)] = ?

Answer 3

cos(x)

Question 4

d/dx[cos(x)] = ?

Answer 4

-sin(x)

Question 5

d/dx[tan(x)] = ?

Answer 5

sec^2(x)

Question 6

d/dx[cot(x)] = ?

Answer 6

-csc^2(x)

Question 7

d/dx[sec(x)] = ?

Answer 7

sec x tan x

Question 8

d/dx[csc(x)] = ?

Answer 8

-csc x cot x

Question 9

A particle is shot upward with an initial velocity of 32 ft/sec from a height of 48 ft.

Answer 9

1) Give the equation which models its position
2) Find the maximum height the object reaches
3) Find the velocity of the object when it hits the ground.

Question 10

Give the equation which models its position

Answer 10

s(t) = -16t^2 + 32t + 48

Question 11

Find the maximum height the object reaches

Answer 11

v(t) = s’(t) = -32t + 32
Set -32t + 21 = 0
32 = 32t
1 = t
s(1) = 16(1)^2 + 32(1) + 48
= 64 feet

Question 12

Find the velocity of the object when it hits the ground.

Answer 12

-16t^2 + 32t + 48 / -16 = 0 / -16
t^2 - 2t - 3 = 0
(t - 3) (t + 1) = 0
t = 3
v(3) = -32(3) + 32
= -64 ft/sec