Question

Let f(x) = x*g(h(x)) where g(4) = 2, g'(4) = 3, h(3) = 4, and h'(3) = -2. Find f'(3).

Answer 1

By the product and chain rules,

f'(x) = g(h(x)) + x g'(h(x)) h'(x)

Then

f'(3) = g(h(3)) + 3 g'(h(3)) h'(3)

f'(3) = g(4) + 3 g'(4) (-2)

f'(3) = g(4) - 6 g'(4)

f'(3) = 2 - 6(3)

f'(3) = -16