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.) We want , where . First, we'll need to find  and so we'll take the derivative of with respect to :

Note the use of the chain rule at . Now, we need to evaluate at 1, so:

b.) We want , where . Again, we'll need :

Note the use of the product rule at . All that's left is to evaluate at ; thus: