Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 4 / a / Solution

The fundamental theorem of calculus differentiates the upper endpoint, while differentiation under the integral sign differentiates the parameter:These formulas hold, for example, when $g$ and its displayed parameter partial derivative are continuous on a neighborhood of the integration interval and the parameter value.