Applying Leibniz's Rule to Differentiate Integrals

[mathmadx]

Dear all, a question which has puzzled me for some days:
(Assume that all are differentiable enough times):

Calculate:
<br /> \frac{\mathrm{d} }{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt<br />

 

[HallsofIvy]

Dear all, a question which has puzzled me for some days:
(Assume that all are differentiable enough times):

Calculate:
<br /> \frac{\mathrm{d} }{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt<br />

Leibniz's rule, which generalizes the fundamental theorem of Calculus:
\frac{d}{dx} \int_{g(x)}^{h(x)} f(x,t)dt= \frac{dh}{dx}f(x, h(x))- \frac{dg}{dx}f(x,g(x))+ \int_{g(x)}^{h(x)} \frac{\partial f(x,t)}{\partial x} dt