Derivative of function with respect to function

Question

Is this acceptable notation: $$\frac{df(x)}{dg(x)} = \lim_{\Delta x \to 0}\frac{f(\Delta x + x)-f(x)}{g(\Delta x + x) - g(x)}$$ I read various opinions in this regard. Sometimes people write $$\frac{df(x)}{d(g(x))}$$ or $$ df /dg.$$ Let me be a bit more specific. In my work, $g(x) = ln(x)$. So the expression directly above would become $$ df /dln .$$ Obviously, that is unacceptable. Suggetions please.

Answer 1

I suppose this is a question of how you introduce the chain rule.

Technically we differentiate with respect to variables, and those variables are related by functions, and not with respect to the function itself.

So, I suppose the correct thing to do is to introduce a variable $u = g(x)$ and then say $\frac {d}{du} f(u) = \frac {df}{du}.$ And, $\frac {d}{dx} f(u) = \frac {df}{du}\frac {du}{dx}.$ But, this seems cumbersome and so it is not uncommon to write $\frac {df}{dg}\frac {dg}{dx}$ where $g$ is a variable $g = g(x).$ But, you can think of $g$ as the function $g(x)$

If we use the Lagrangian (primed) notation we would say $(f\circ g)'(x) = f'(g(x))g'(x)$

This is standard notation even if you do not like the aesthetics of the $f'(g(x))$ factor. That factor is the same factor as $\frac {df}{dg}$ in the Liebnitz notation.

As for $\frac {df}{d\ln x}$ (or $\frac {df}{d\ln x}\frac {d\ln x}{dx}$), that looks ugly to me, and would rather have a variable in the operator and not a function. But, then I don't particularly care for a function in the top of the differential operator e.g. $\frac {d \sin x}{dx} = \cos x$ either, even though that is a pretty common notation. I would rather write $\frac {d}{dx} \sin x = \cos x$