Chain Rule Question?

Question

Let f be a differentiable function and define f2(x)= f(f(x)), f3(x)= f(f(f(x))) and so on. Find the expression for the derivative of fn(x), where n is any integer greater than or equal to 2.

I don't understand how to put this in terms of n! Please help!

Answer 1

Start off with f2. Its derivative is:

f'(f(x)*f'(x)

Then look at f3. Its derivative is:

f'(f2(x))*f'(f(x)*f'(x)

In general, the derivative of fn(x) will be:

f'(f(x))*f'(f(x))* . . . f'(f2(x))*f'(f(x)*f'(x)

You can prove it by induction if you need a proof.