# Leibniz's Integral Rule

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## 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 $\tuple {x, t} \in R$:

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

## Proof

## 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:

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

## Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 15$: Leibnitz's Rule for Differentiation of Integrals: $15.14$

- Weisstein, Eric W. "Leibniz Integral Rule." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/LeibnizIntegralRule.html