Question understanding the chain rule

Question

I have that $f(z)$ is differentiable for all z in the complex plane. $g(z)$ = $f(z^2)$ and $g(z)$ has the form $g(z)$ = $u(x,y)$ + $iv(x,y)$. I'm trying to find $g'(z)$ and I know I need to use the chain rule. My question is will this become $f'(z^2)$*$2z$ or would it be $f'(2z)$ * $f(2z)$?

Answer 1

Set $h(z) = z^2$, then $g(z) = f(h(z))$ and chain rule says $$ g'(z) = (f\circ h)'(z) = f'(h(z))h'(z)$$ so the answer is $f'(z^2)2z$.