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)).

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