Also known as Feynman's trick or differentiation under the integral sign.
In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form $$ \int_{a(x)}^{b(x)}f(x,t)\, dt \text , $$ where $ -\infty < a(x),b(x)< \infty $, the derivative of this integral is expressible as $$\frac{d}{dx}\int_{a(x)}^{b(x)}f(x,t)\, dt=f\big(x,b(x)\big)b'(x)-f\big(x,a(x)\big)a'(x)+\int_{a(x)}^{b(x)}\frac{\partial}{\partial x}f(x,t)\,dt\text.$$ In particular,
$$\frac{d}{dx}\int_a^bf(x,t)\, dt=\int_{a}^{b}\frac{\partial}{\partial x}f(x,t)\,dt.$$
