We study a variational principle in which there is one common perturbation function $\vv$ for every proper lower semicontinuous extended real-valued function $f$ defined on a metric space $X$. Necessary and sufficient conditions are given in order for the perturbed function $f+\vv$ to attain its minimum. In the case of a separable Banach space we obtain a specific principle in which the common perturbation function is, in addition, also convex and Hadamard-like differentiable. This allows us to provide applications of the principle to differentiability of convex functions on separable and more general Banach spaces.