Derivative of an integral

Let f be a Riemann integrable function on an interval [a,b]. Fix some c from this interval and define

By the Fundamental Theorem of Calculus, F ′ = f. However, very often the upper limit in the integral is not just x, but some function of it. That is, we take some differentiable function g on [a,b] with range contained in [a,b] and define

What is the derivative of F now? To get the answer, we define

By the Fundamental Theorem of Calculus, G′(y) = f (y). The function F is a composition of G(y) and y = g(x), by the chain rule we therefore easily get

This can be also expressed like this:

Gamma and Beta functions
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