In calculus, the Leibniz integral rule or the Leibniz rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form ∫ a ( x ) b ( x ) f ( x , t ) d t , {\displaystyle \int _{a(x)}^{b(x)}f(x,t)\,dt,}
where − ∞ < a ( x ) , b ( x ) < ∞ {\displaystyle -\infty <a(x),b(x)<\infty } and the integrands are functions dependent on x , {\displaystyle x,} the derivative of this integral is expressible as
d d x ( ∫ a ( x ) b ( x ) f ( x , t ) d t ) = f ( x , b ( x ) ) ⋅ d d x b ( x ) − f ( x , a ( x ) ) ⋅ d d x a ( x ) + ∫ a ( x ) b ( x ) ∂ ∂ x f ( x , t ) d t {\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 ∂ ∂ x {\displaystyle {\frac {\partial }{\partial x}}} indicates that inside the integral, only the variation of f ( x , t ) {\displaystyle f(x,t)} with x {\displaystyle x} is considered in taking the derivative. In the special case where the functions a ( x ) {\displaystyle a(x)} and b ( x ) {\displaystyle b(x)} are constants a ( x ) = a {\displaystyle a(x)=a} and b ( x ) = b {\displaystyle b(x)=b} with values that do not depend on x , {\displaystyle x,} this simplifies to:
d d x ( ∫ a b f ( x , t ) d t ) = ∫ a b ∂ ∂ x f ( x , t ) d t . {\displaystyle {\frac {d}{dx}}\left(\int _{a}^{b}f(x,t)\,dt\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,t)\,dt.}
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