Leibniz's Integral Rule

Theorem
Let $f \left({x, t}\right)$, $a \left({t}\right)$, $b \left({t}\right)$ be continuously differentiable functions on some region $R$ of the $\left({x, t}\right)$ plane.

Then for all $\left({x, t}\right) \in R$:


 * $\displaystyle \frac {\mathrm d} {\mathrm d t} \int_{a \left({t}\right)}^{b \left({t}\right)} f \left({x, t}\right) \, \mathrm d x = f \left({t, b \left({t}\right)}\right) \frac {\mathrm d b} {\mathrm d t} - f \left({t, a \left({t}\right)}\right) \frac {\mathrm d a} {\mathrm d t} + \int_{a \left({t}\right)}^{b \left({t}\right)} \frac {\partial} {\partial t} f \left({x, t}\right) \, \mathrm d x$

If $a \left({t}\right)$ and $b \left({t}\right)$ are constant, i.e $a \left({t}\right) = a$ and $b \left({t}\right) = b$ for some $a, b \in \R$, then we can write this as:


 * $\displaystyle \frac {\mathrm d} {\mathrm d t} \int_a^b f \left({x, t}\right) \, \mathrm d x = \int_a^b \frac {\partial} {\partial t} f \left({x, t}\right) \, \mathrm d x$