Leibniz's Integral Rule

Theorem
Let $\map f {x, t}$, $\map a t$, $\map b t$ be continuously differentiable real functions on some region $R$ of the $\tuple {x, t}$ plane.

Then for all $\tuple {x, t} \in R$:


 * $\ds \frac \rd {\rd t} \int_{\map a t}^{\map b t} \map f {x, t} \rd x = \map f {\map b t, t} \frac {\rd b} {\rd t} - \map f {\map a t, t} \frac {\rd a} {\rd t} + \int_{\map a t}^{\map b t} \frac {\partial} {\partial t} \map f {x, t} \rd x$

Also known as
This is also referred to in some sources as Leibniz's Rule, but as this name is also used for a different result, it is necessary to distinguish between the two.

Other popular names for this technique include differentiation under the integral sign and Feynman's technique after physicist Richard Feynman.

Also see

 * Definite Integral of Partial Derivative, where $\map a t$ and $\map b t$ are constant:


 * $\ds \frac {\rd} {\rd t} \int_a^b \map f {x, t} \rd x = \int_a^b \frac {\partial} {\partial t} \map f {x, t} \rd x$