imagine you have a function $f(x)$, which is non-zero on an intervall $[X_1,X_2]$ and zero everywhere else. It is continiuos at $X_1$. You have another function $g(x)$, which is non-zero $[X_0, X_1]$ and drops down to zero after $X_1$, and is unsteady there. Here's the kicker: $X_1$ is a function of $g(x)$ as in $X_1$ is defined as the x-coordinate where $g(x)$ reaches a certain $y_1=g(X_1)$. I.e. the function $f(x)$ is triggered to start and be non-zero once $g(x)$ hits a certain value.
I am now looking for the partial derivative
$\frac{\partial f(x)}{\partial g(x)}$.
Both $\frac{\partial f(x)}{\partial X_1}$ and $\frac{\partial X_1}{\partial g(x)}=\frac{\partial g(x)}{\partial X_1}^{-1}$ can be obtained, I think, but using the chain-rule seems very wrong?
Also, what would
$\int_{X_0}^{X_2}\frac{\partial f(x)}{\partial g(x)}$.
look like? Are any of these expressions even defined? Send help :(