Leibniz integral rule
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 .