In mathematics, there is a formula for differentiation under the integral sign in calculus.

Assume

where a(x), b(x), f(x,t) are differentiable with respect to x functions. Then

This can be proven using the fundamental theorem of calculus. Incidentally, the fundamental theorem of calculus is a particular case of the above formula, for a(x) = a a constant, b(x) = x, f(x,t) = f(t).

Another case of interest is to take both upper and lower limits constant. Then the formula takes the shape of an operator equation

I_tD_x = D_xI_t

where D_x is the partial derivative with respect to x and I_t is the integral operator with respect to t over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as Leibniz's rule (derivatives and integrals).