I consider a bivariate function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $f(x,\cdot)$ is strictly convex for any $x$. The strict convexity implies that
$$ y^*(x) = \arg \min_{y \in \mathbb{R}} f(x,y) $$
is uniquely defined for any $x \in \mathbb{R}$. I am curious to know if formulas for the derivatives of $y^*$ exists (together with conditions on $f$ ensuring the existence of derivatives of $y^*$). The problem seems very natural but I could not find easily some elementary or introductive answer to my question.
I am also interested in multivariate or even infinite-dimensional generalizations.