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.

Also see

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


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