Partial derivative of function of inverse function

Question

I have got a probelm with the following task: $\frac{\partial }{\partial x}f(f^{-1}(x,t),\tau)$, where $f\in\mathscr{C}^{\infty}(\mathbb{R^2})$. My attemp is $\frac{\partial }{\partial x}f(f^{-1}(x,t),\tau)=\frac{\partial }{\partial x}f(f^{-1}(x,t),\tau)\frac{\partial }{\partial x}f^{-1}(x,t)$, but I am not sure, if it holds. Many thanks for any hints.

Answer 1

Unless I have misinterpreted your notation, the problem does not make sense. If $f$ is a function with domain $\mathbb{R}^2$, and $f^{-1}$ is supposed to be its inverse, then the codomain of $f^{-1}$ must be $\mathbb{R}^2$. But then it does not make sense to write

$$f(f^{-1}(x,t),\tau).$$

To elaborate: $f^{-1}(x,t)\in\mathbb{R}^2$ and so ($f^{-1}(x,t),\tau)\in\mathbb{R}^2\times\mathbb{R}$. This cannot be an argument of $f$ because $f$ only takes elements of $\mathbb{R}^2$ as arguments.