“Differentiating under the Integral” is a useful trick, and here we describe and prove a sufficient condition where we can use the trick. This is the Measure-Theoretic version, which is more general than the usual version stated in calculus books.
Let be an open subset of , and be a measure space. Suppose satisfies the following conditions:
1) is a Lebesgue-integrable function of for each .
2) For almost all , the derivative exists for all .
3) There is an integrable function such that for all .
Then for all ,
Let be a sequence tending to 0, and define
It follows that is measurable.
Using the Mean Value Theorem, we have for each .
Thus for each , by the Dominated Convergence Theorem, we have which implies