Leibnitz Integral Rule

The Leibnitz Rule’s goal is to provide a formula for differentiating an integral whose limits and/or integrand depend on a variable. The rule for the derivative of an integral having limits $a(t)$ and $b(t)$ is stated as

$$\frac{d}{dt}\int \_{a(t)}^{b(t)}f(t,x)dx=\int \_{a(t)}^{b(t)}\frac{df}{dt}dx-f(a,t)\frac{da}{dt}+f(b,t)\frac{db}{dt}$$