Consider the following problem,
\begin{align} c(y,\mathbf{w})=\inf_{\substack{\mathbf{x} \in \mathbb{R}^n_{+} \\ \text{s.t. }f(\mathbf{x}) \geq y }} \mathbf{w} \cdot \mathbf{x} \end{align}
where $c(y,\mathbf{w})$ is the value function with $c: \mathbb{R}^n_{++} \times \mathbb{R}_+ \rightarrow \mathbb{R}_+$(or equivalently, given $y$, it is the support function of the set $\left\{\mathbb{x} \in \mathbf{R}^n_{+}: f(\mathbb{x}) \geq y \right\}$).
Suppose further that $f(\cdot)$ is continuous, strictly increasing and strictly quasi-concave and $f(0)=0$. Now I have a theorem that states that $c(y,\cdot)$ is differentiable for any $y \in \mathbb{R}_+$, at any $\mathbf{w} \in \mathbb{R}^n_{++}$ under these circumstances.
How can I prove that $c(\cdot,\cdot$) is differentiable, i.e. $Dc(y,w)$ exists. I know that proving the differentiability of $c(\cdot,\mathbf{w})$ for any $\mathbf{w}$ is insufficient, since a multi-variable function can be non-differentiable (Jacobian fails to exist), despite being differentiable in individual arguments.
Any help is appreciated, thanks