Leibniz integral rule

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

then for ${\displaystyle -\infty <a(x),b(x)<\infty }$ the derivative of this integral is expressible as

where the partial derivative indicates that inside the integral, only the variation of f(•,t) with t is considered in taking the derivative. Notice that if ${\displaystyle a(x)}$ and ${\displaystyle b(x)}$ are constants rather than functions of ${\displaystyle x}$, we have a special case of Leibniz's rule:

...
Wikipedia