6. 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)

1 3 2 4 6

2 1 8 5 7

3 7 2 7 9

a. Let 𝐻(𝑥) = 𝑓(𝑔(𝑥)). Find 𝐻 ′ (1).

b. Let 𝑅(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥). Find 𝑅′(2).

Answer 1

a. H(x) is defined as f(g(x)). To find H'(1) we must find the derivative of H(x). To do this, we'll need to use chain rule:

H'(x) = f '(g(x)) * g'(x)

Now that we have H'(x), we can solve for H'(1):

H'(1) = f '(g(1)) * g'(1)

= f '(2) * 6

= 5 * 6

= 30

b. R(x) is defined as f(x) * g(x). To find R'(2) we must find the derivative of R(x). To do this, we'll need to use product rule:

R'(x) = f(x) * g'(x) + f '(x) * g(x)

Now that we have R(x), we can solve for R'(2):

R'(2) = f(2) * g'(2) + f '(2) * g(2)

= 1 * 7 + 5 * 8

= 47