Derivatives of integration over level sets of a function

Question

I want to take derivatives of the function given by the integral: $$ \int_{\{x\in\mathbb{R}^n: f=t\}}g(x)dx $$ with respect to t. Someone told me there is a proposition in manifold and geometry that such kind of derivatives have something to do with the normal vector fields. Could someone kindly help me with this kind of derivation problem? Thank you!