differentiation of multivalued function integration

Question

I wonder about differentiation of integrated multivalued function such as $\frac{d}{dt}\int_0^tf(x,t)dx$. Is there any well-known formula to slove it?

Answer 1

You can use the Leibniz integral rule of differentiation under the integral sign

$\small\,\frac{d}{dx}\bigg(\large \int_{\small a(x)}^{ b(x)}\large f(x,t)\,dt\bigg) =\small f(x,b(x)).\frac{d(b(x))}{dx}-f(x,a(x)).\frac{d(a(x))}{dx}+\int_{\small{a(x)}}^{\small{b(x)}}\partial_xf(x,t)dt$

In your case it becomes,

$f(t,t)+\int_0^t\partial_tf(x,t)\,dx$

Answer 2

You can calculate it this way: $$g(\alpha,\beta)=\int_0^{\alpha} f(x,\beta)dx$$ $$\frac{d}{dt}g(t,t)=(\partial_{\alpha}g)(t,t) + (\partial_{\beta}g)(t,t)$$ $$\frac{d}{dt}g(t,t)=f(t,t)+\int_0^{t} (\partial_{\beta}f)(x,t)dx$$