Most of the calculus I've studied seems separate math problems in to "derivative" or differential applications and integral applications.
The one exception seems to be "calculus of variations," which seems to combine both by taking integrals of differential equations and partial derivatives.
Richard Feynman's autobiography included the quote, "differentiat[ing] parameters under the integral sign...is not taught very much in the universities; and I used that one damn tool again and again...So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me."
Is it true that "differentiating under the integral sign" is one of the keys to learning calculus of variations?
No, in fact differentiating under the integral sign refers to something different: A technique of carrying out tricky integrals. One introduces an additional parameter in the integrand which somehow induces a simplification, allowing one to solve a more general but simpler problem, after which one simply sets the additional, artificially introduced parameter to the value which corresponds to the original integral one was trying to solve in the first place. For some examples and more background information, you can check out this useful PDF or the appropriate wikipedia page.