Calculus notation: is $\frac{dy}{dx}$ the same as $y'$ if $y\not = y(x)$?

Question

Suppose $y=y(ax)$ for some $a\in \mathbb{R}$. Would I correct in stating that $\frac{dy}{dx}\not =y'(ax)$ but rather the following is true: $$\frac{dy}{dx}=ay'(ax)=y'(x)?$$ Hence $\frac{dy}{dx}=y'(x)$ is still true as long as we say the argument for $y'$ is still in $x$?