Writing The Derivative Of $f(x)$ With Respect To $g(x)$ In Limit Form

Question

What would be the proper way to represent this derivative in the limit form? $$\frac{\mathrm{d} }{\mathrm{d} g(x)}[f(g(x))]$$

In my attempt to solve this I've tried to word out the derivative: The "infinitely small" change in $g(x)$ under the corresponding "infinitely small" change in $f(g(x))$.

Making sense of this sentence, I was able to come up with this limit: $$\lim_{\Delta g(x)\rightarrow 0}\frac{f(g(x) + \Delta g(x)) - f(g(x))}{\Delta g(x)}$$

Is this correct, and is it ok to be using $\Delta g(x)$ as my "value under the lim"?

Answer 1

You are right.

I perfer to write in this form:

$$\lim_{\Delta U\rightarrow 0}\frac{f(U + \Delta U) - f(U)}{\Delta U}$$

Where $f(g(x))=f(U)$, and $U=g(x)$.