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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