So I've seen several questions asking how one differentiates with respect to a function. I was curious what exactly this means, or what exactly the limit quotient we're using here is.
People seem to just expand $\frac{df}{dg} = \frac{df}{dx} \frac{dx}{dg}$, which may be fine for some applications, but I was curious if the quantity $\frac{df}{dg}$ had a more formal definition.
Is just that if $f(t) = h(t, g)$ with $g = g_1(t)$ (assuming the injectivity and domains work out), we substitute $t = g^{-1}(g)$, and then differentiate $f(g) = h(g^{-1}_1 (g), g)$ with respect to g?
Thanks.
I think that it helps to be careful and not to use the same letter for different objects: at least when trying to clarify what is going on.
The careful story is this.
We have two functions, $f,g:\mathbb{R}\to\mathbb{R}$.
From these we can define a new function, $F:\mathbb{R}\to\mathbb{R}$ by $F(x):=f(g(x))$.
By the Chain Rule we then have $F'(x)=f'(g(x))g'(x)$.
In $\frac{d}{dx}$-language this is written $\frac{dF}{dx}=\frac{df}{dg}\frac{dg}{dx}$; and for those who don't distinguish between $f$ and $F$ (after all they are different expressions for the same quantity) this will be $\frac{df}{dx}=\frac{df}{dg}\frac{dg}{dx}$.