Derivative of a function which is treated as a variable

Question

I have got a function $f=f(x)$. The derivative is $\partial_xf$. There are applications in which it is reasonable to treat $f$ as another variable in a larger context. In my application I now need an expression for the derivative with respect to $f$ of $\partial_xf$, i.e. $\partial_f(\partial_xf)$. I don't know much about operator calculus. Does $$\partial_f(\partial_xf)=\frac{\partial}{\partial f}\frac{\partial f}{\partial x}\overset{?}{=}\frac{\partial}{\partial x}=\partial_x$$ hold? Is $\partial_x$ the correct expession? If yes: How can it be shown? If not: What is the right one?