1. Assume that 𝑓(𝑥) and 𝑔(𝑥) are differentiable functions and all we know about them is given in the table shown below.

Question

X F(x) G(x) F'(x) G'(x)

-2 3 1 -5 8

-1 -9 7 4 1

0 5 9 9 -3

1 3 -3 2 6

2 -5 3 8 0

a. Let 𝑝(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥). Find 𝑝 ′ (1).

b. Let ℎ(𝑥) = 𝑔(𝑥) ∙ sin(𝑥). Find ℎ′(0).

c. Let 𝑗(𝑥) = [𝑓(𝑥) + 𝑥 2 ] ^3 . Find 𝑗′(1)

d. Let 𝑞(𝑥) = sin(𝑥)/𝑔(𝑥) . Find 𝑞′(0).

Answer 1

a) p '(x) = f '(x)g(x) + f(x)g '(x) , p '(1) = 2(-3)+3(6) = 12

b) h'(x) = g'(x)sin(x) + g(x)cos(x), p'(0) = 0 + (-3) = -3

c) j '(x) = 3(f(x)+ x²)²(f '(x) + 2x), f '(1) = 3(2)²(2+2) = 48

d) q'(x) = (cos(x)g(x)-sin(x)g '(x))/g²(x), q '(0) = (1(-3)-0)/(-3)² = -3