By the way, here is the general Leibniz formula:

\(\displaystyle \frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} F(x,t)dt= \)\(\displaystyle F(x, \beta(x))\frac{d\beta(x)}{dx}- F(x,\alpha(t))\frac{d\alpha(x)}{dx}+\)\(\displaystyle \int_{\alpha(x)}^{\beta(x)}\)\(\displaystyle \frac{\partial F(x,t)}{\partial x} dt\).

It can be proved from the fundamental theorem of calculus and using the chain rule for both the upper and lower limits.