Question

If​ f(x) and​ g(x) are differentiable functions such that f (1 )equals 4​, f prime (1 )equals 5​, f prime (9 )equals 6​, g (1 )equals 9​, g prime (1 )equals 8​, g prime (2 )equals 9​, g prime (5 )equals 10​, find StartFraction d Over dx EndFraction f (g (x ))vertical line Subscript x equals 1 Superscript .

Answer 1

By the chain rule,

$(f(g(x)))'=f'(g(x))g'(x)$

so that when $x=1$, we have

$(f(g(1)))'=f'(g(1))g'(1)=f'(9)g'(1)=6\cdot8=\boxed{48}$