This concept covers differentiation under the integral sign, a technique in multivariable calculus for finding the derivative of a function defined as a definite integral whose integrand (and possibly its limits of integration) depend on a parameter. When the parameter appears only in the integrand, the derivative is obtained by moving the differentiation operator inside the integral and replacing it with the partial derivative of the integrand with respect to the parameter, provided the integrand and its partial derivative are continuous so that the interchange of the limit and integration operations is valid. When the parameter also determines the limits of integration, the result generalizes via the chain rule (Leibniz's rule), combining the interior partial-derivative term with boundary terms accounting for the rate of change of the upper and lower limits. This topic belongs to real analysis/multivariable calculus and connects the theory of uniform convergence (interchange of limit and integral) with the chain rule for functions of several variables, with applications in physics, probability theory, and integral equations.
This concept covers differentiation under the integral sign, a technique in multivariable calculus for finding the derivative of a function defined as a definite integral whose integrand (and possibl…