Question

If f is the function whose graph is shown, let h(x) = f(f(x)) and g(x) = f(x2).
I am having problems with g'(2). 

Answer 1

By the chain rule,

$g(x)=f(x^2)\implies g'(x)=2xf'(x^2)$

so

$g'(2)=2\times2\times f'(2^2)=4f'(4)$

You were able to find $h'(2)=f'(f(2))\times f'(2)$, which requires knowing both $f(2)$ and $f'(2)$. Do you also happen to know $f(4)$ and $f'(4)$?