Derivative notation?

Question

I am getting a bit confused with the primed notation for derivatives, does $$f'(g(x))$$ mean the first derivative of $f$ with respect to the spacial coordinate $x$ or with respect to $g(x)$. If it is the latter case, then, does the ' notation all ways represent the derivative with respect to the argument, in this case $g(x)$?

Answer 1

Note that $$f'(g(x))=f'(u)\big|_{u=g(x)}$$

Answer 2

If you want to take the derivative w.r.t. $x$ write either $(f(g(x)))^\prime$ (uncommon) or $(f\circ g)^\prime (x)$ or $\frac{d}{dx}f(g(x)) $ or $\frac{d}{dx}f\circ g(x)$. What you wrote down is what Paul explained, the derivative of $f$ evaluated at $g(x)$.