Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows: Theorem. Let f(x, t) be a function such that both f(x, t) and its partial derivative fx(x, t) are continuous in t and x in some region of the (x, t)-plane, including a(x) ≤ t ≤ b(x), x0 ≤ x ≤ x1. Also suppose that the functions a(x) and b(x) are both continuous and both have continuous derivatives for x0 ≤ x ≤ x1. Then for x0 ≤ x ≤ x1: This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The [second] fundamental...