Corollary

If the partial derivative $\mathbf{f}'_{\alpha}(x, \alpha)$ exists on $I \times V$ and is a continuous function of $(x, \alpha)$ on this set, then the function $g$ admits a derivative given by the formula (12) at the point $\alpha_0$.

Indeed, if W is a compact neighbourhood of $\alpha_0$ contained in V, then the map $(x, \alpha) \mapsto \mathbf{f}'_{\alpha}(x, \alpha)$ is uniformly continuous on the compact set $I \times W$, so $\mathbf{f}'_{\alpha}(x, \alpha)$ tends to $\mathbf{f}'_{\alpha}(x, \alpha_0)$ uniformly on I as $\alpha$ tends to $\alpha_0$.

From prop. 5 one deduces a more general proposition which allows one to evaluate the derivative of an integral when, not only the integrand $\mathbf{f}$, but also the limits of integration, depend on a parameter $\alpha$: