Leibniz Integral Rule

The Leibniz integral rule for differentiation under the integral sign:

General Case

$$\begin{array}{c}{\int }_{a(x)}^{b(x)}f(x,t)dt\\ \\ a(x),b(x)\in \mathbb{R}\\ \\ \downarrow \\ \\ \frac{d}{dx}\left({\int }_{a(x)}^{b(x)}f(x,t)dt\right)\\ ||\\ f(x,b(x))b^{\prime }(x)-f(x,a(x))\cdot a^{\prime }(x)+{\int }_{a(x)}^{b(x)}\frac{\partial }{\partial x}f(x,t)dt\end{array}$$

Variable Upper Bound

$$\begin{array}{c}{\int }_a^{x}f(x,t)dt\\ \\ \downarrow \\ \\ \frac{d}{dx}\left({\int }_a^{x}f(x,t)dt\right)\\ ||\\ f(x,x)+{\int }_a^x\frac{\partial }{\partial x}f(x,t)dt\end{array}$$

Constant Bounds

$$\begin{array}{c}{\int }_a^{b}f(x,t)dt\\ \\ \downarrow \\ \\ \frac{d}{dx}\left({\int }_a^{b}f(x,t)dt\right)\\ ||\\ {\int }_a^b\frac{\partial }{\partial x}f(x,t)dt\end{array}$$