Question

Given f(1) = 2, f′(1) = 3, g(1) = 1, g′(1) = 5, compute the following values: (a) Compute h′(1) for h(x) = f(g(x)). ′1 (b) Compute j (1) for j(x) = f(x). (c) Compute k′(1) for k(x) = ln(g(x)).

Answer 1

a. By the chain rule,

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

h'(1) = f'(g(1)) * g'(1) = f'(1) * 1 = 3

b. I suspect there's a typo here somewhere, but if you really mean j(x) = f(x), and you're only supposed to find j(1), then

j(1) = f(1) = 2

Possibly you're supposed to instead find j'(1), in which case

j'(1) = f'(1) = 3

Or maybe j is defined like

j(x) = 1/f(x)

in which case the chain rule gives

j'(x) = -f'(x)/f(x)^2

j'(1) = -f'(1)/f(1)^2 = -3/2^2 = -3/4

c. By the chain rule,

k'(x) = g'(x)/g(x)

k'(1) = g'(1)/g(1) = 5/1 = 5