Leibniz integral rule

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In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form $${\displaystyle \int _{a(x)}^{b(x)}f(x,t)\,dt,}$$ where ${\displaystyle -\infty <a(x),b(x)<\infty }$ and the integrands are functions dependent on ${\displaystyle x,}$ the derivative of this integral is expressible as $${\displaystyle {\begin{aligned}&{\frac {d}{dx}}\left(\int _{a(x)}^{b(x)}f(x,t)\,dt\right)\\&=f{\big (}x,b(x){\big )}\cdot {\frac {d}{dx}}b(x)-f{\big (}x,a(x){\big )}\cdot {\frac {d}{dx}}a(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}f(x,t)\,dt\end{aligned}}}$$ where the partial derivative ${\displaystyle {\tfrac {\partial }{\partial x}}}$ indicates that inside the integral, only the variation of ${\displaystyle f(x,t)}$ with ${\displaystyle x}$ is considered in taking the derivative.^[1]

In the special case where the functions ${\displaystyle a(x)}$ and ${\displaystyle b(x)}$ are constants ${\displaystyle a(x)=a}$ and ${\displaystyle b(x)=b}$ with values that do not depend on ${\displaystyle x,}$ this simplifies to: $${\displaystyle {\frac {d}{dx}}\left(\int _{a}^{b}f(x,t)\,dt\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,t)\,dt.}$$

If ${\displaystyle a(x)=a}$ is constant and ${\displaystyle b(x)=x}$, which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes: $${\displaystyle {\frac {d}{dx}}\left(\int _{a}^{x}f(x,t)\,dt\right)=f{\big (}x,x{\big )}+\int _{a}^{x}{\frac {\partial }{\partial x}}f(x,t)\,dt,}$$

This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and 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.