Can you find a derivative of a function $f(x)$ with respect to something that isn't $x$?

Question

I am not sure how to phrase this exactly, but an example of what I'm talking about is finding the derivative of $x^4$ with respect to, say, $x^2$. I was just thinking, maybe you could use some substitution to find the answer, and make $x^2=a$, and thus find $d/dx$ of $a^2$ with respect to $a$, and hence $d/d(x^2)$ of $x^4$ would be $2x^2$. Can you do this? Or is it illegal?

Answer 1

Let $y=x^2.$

Then, by the inverse function rule, $$\frac{\mathrm dx}{\mathrm dy}=\frac1{\frac{\mathrm dy}{\mathrm dx}}.$$

By the chain rule, \begin{align}\frac{\mathrm d}{\mathrm dy}\left(x^4\right)&=\frac{\mathrm d}{\mathrm dx}\left(x^4\right)\cdot\frac{\mathrm dx}{\mathrm dy}\\&=4x^3\cdot\frac1{2x}\\&=2x^2.\end{align}