Question

If
F(x) = f(xf(xf(x))),
where
f(1) = 4, f(4) = 6, f '(1) = 4, f '(4) = 5,
and
f '(6) = 6,
find
F '(1).

Answer 1

Let's take this a step at a time

F' = f'(xf(xf(x))) * (xf(xf(x)))'
(xf(xf(x)))' = f(xf(x)) + xf'(xf(x))
f'(xf(x)) = f(x) + xf'(x)

so:

F'(1) = f'(1f(1f(1))) * (f(1f(1)) + f(1)+f'(1))
= f'(f(4)) * (f(4)+f(1)+f'(1))
= f'(6) (6+4+4)
= 6*14
= 94

Better double-check it, since I don't seem to have used f'(4)=5

Answer 2

94 is not right