Let $\zeta: \mathbb{R^m} \times \mathbb{R^n} \mapsto \mathbb{R}$ be a smooth function and define $\phi(x) = \inf_{y \in C} \zeta(x,y)$ where $C \subset \mathbb{R^n}$ is compact. Suppose that for every $x \in X \subset \mathbb{R^m}$ there is a unique $y(x) \in C$ such that $\phi(x) = \zeta(x,y(x))$.
Now, according to a book a have, $\frac{\partial \phi}{\partial x}(x_0) = \frac{\partial \zeta}{\partial x} (x_0, y(x_0)) \quad \forall x_0 \in X$ (this is let as a exercise). The problem is: 1) this looks simple but I don't where to start to prove this, 2) even if I knew how to deal with the first part, I also need the second differential (the Hessian) of $\phi$ (given in term of $\zeta$, $y$ and their derivatives).
I tried to do something like $\frac{\partial \phi}{\partial x}(x_0) = \frac{\partial \zeta}{\partial x} (x_0, y(x_0)) + \frac{\partial \zeta}{\partial y} (x_0, y(x_0)) \frac{\partial y}{\partial x}(x_0)$ (this suggests that $\frac{\partial y}{\partial x}(x_0) = 0$) and find the derivative of $y$ by the implicit function theorem, but I can only get the tautology $\frac{\partial \phi}{\partial x}(x_0) =\frac{\partial \phi}{\partial x}(x_0)$, so I suspect this isn't the good way.
Note: this question is related to this one (Infimum is a continuous function, compact set).