Suppose I have a mix of a differential operator and a function in the form $f(x)\frac{d}{dx}$, which may be applied to a function $g(x)$ to make $f(x)\frac{d}{dx}g(x) = f(x)g'(x)$.
However, what if I don't want to deal with $f(x)$, and I just want to find a single derivative with respect to some other coordinate system?
Then does there exist a substitution such that $f(x) \frac{d}{dx} = \frac{d}{du}$? And if so, what effect does such a transformation have on $g(x)$?
In any interval where $f$ does not vanish we have $f(x)\frac d {dx} =\frac d {du}$ where $u(x)=\int_c^{x} \frac 1 {f(t)} dt$ where $c$ is a fixed point in the interval. [To verify this just note that $\frac {du} {dx} =\frac 1 {f(x)}$].