# Leibniz integral rule

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

${\displaystyle \int _{a(x)}^{b(x)}f(x,t)\,dt,}$

where ${\displaystyle -\infty , the derivative of this integral is expressible as

${\displaystyle {\frac {d}{dx}}\left(\int _{a(x)}^{b(x)}f(x,t)\,dt\right)=f{\big (}x,b(x){\big )}\cdot {\frac {d}{dx}}b(x)-f{\big (}x,a(x){\big )}\cdot {\frac {d}{dx}}a(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}f(x,t)\,dt,}$

where the partial derivative indicates that inside the integral, only the variation of ${\displaystyle f(x,t)}$ with ${\displaystyle x}$ is considered in taking the derivative.[1] Notice that if ${\displaystyle a(x)}$ and ${\displaystyle b(x)}$ are constants rather than functions of ${\displaystyle x}$, we have the special case:

${\displaystyle {\frac {d}{dx}}\left(\int _{a}^{b}f(x,t)\,dt\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,t)\,dt.}$

Besides, if ${\displaystyle a(x)=a}$ and ${\displaystyle b(x)=x}$, which is a common situation as well (for example, in the proof of Cauchy's repeated integration formula), we have:

${\displaystyle {\frac {d}{dx}}\left(\int _{a}^{x}f(x,t)\,dt\right)=f{\big (}x,x{\big )}+\int _{a}^{x}{\frac {\partial }{\partial x}}f(x,t)\,dt,}$

Thus under certain conditions, one may interchange the integral and partial differential operators. This important result is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.