Differentiation under the integral sign

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Topics in calculus

Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem

Differentiation

Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives

Integration

Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution

Differentiation under the integral sign is an operation in the mathematical field of calculus.

Assume

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

[edit] Popular culture

• Differentiation under the integral sign is mentioned in the late physicist Richard Feynman's best-selling memoir Surely You're Joking, Mr. Feynman!, (in the chapter "A Different Box of Tools") where he mentions learning it from an old text by Frederick S. Woods' Advanced Calculus (1926) while in high school. The technique was not often taught when Feynman later received his formal education in calculus, and knowing it, Feynman was able to use the technique to solve some otherwise difficult integration problems upon his arrival at graduate school at Princeton University, in part because of the obscurity of the method.