Consider a function $f(x)$.
The derivative of $f$ with respect to $x$ is defined as $$ f'(x)=\lim_{\epsilon\rightarrow 0} \frac{f(x+\epsilon)-f(x)}{\epsilon}. $$
Sometimes, we define a functional dervative of $f$ with respect to $g$ as $$ \frac{\partial f(g(x))}{\partial g(x)}. $$
I think it holds $$ \frac{\partial f(g(x))}{\partial g(x)}=f'(x)|_{x=g(x)}=f'(g(x)). $$ Is it true?? If so, what conditions are required to hold the above relationship? (such as f(x) is differentiable, g(x) is at least continuous.)