I have a function $f : X \times Y \rightarrow\Bbb R$, where $X$ is a convex and compact subset of $R$ and $Y$ is a convex and compact subset of $R^{k}$ for some finite positive integer $k$. (I will be interested in both cases when $k=1$ and when $k>1$.)
Suppose I know: (1) $f$ is continuously differentiable over $X \times Y$; (2) $f( \cdot ,y)$ is strictly increasing for all $y$, so that $f^{-1}( \cdot ,y)$ exists for all $y$.
I would like to know about taking a partial derivative of $f^{-1}( \cdot ,y)$ with respect to $y$. E.g. conditions for which they exist, smoothness properties, and any relation to the derivative of $f$. Any answer or advice on where to look would be very much appreciated!
Your function $(y,z)\longrightarrow x = h(y,z)$ is implicitly defined by $$f(x,y) = z$$ or $$f(h(y,z),y) - z = 0$$ with $$h:Y\times\Bbb R\longrightarrow X.$$ By the implicit function theorem, if $f(x_0,y_0) = z_0$ and $(\partial_x f)(x_0,y_0)\ne 0$, then $h$ will exist near $(y_0,z_0)$ and will be $C^1$. Its partial derivatives will be implicitly determined by partial derivation of the implicit equation:
$$(\partial_x f)(h(y,z),y)\partial_{y_k}h(y,z) + (\partial_{y_k}f)(h(y,z),y) = 0.$$