Differentiate a function under Integral

Question

$$\frac{d}{dt}\int_{a(t)}^{b(t)} g(x;t) dx=g(b(t);t)b'(t)-g(a(t);t)a'(t)+\int_{a(t)}^{b(t)} \frac{\partial}{\partial t} g(x;t)dx$$ How can I get this sentence? My professor said 'just use chain rule', and I figure it out , but not exactly know how this works.

Answer 1

Let $$F(u,v,t)=\int_u^v g(x,t)\,dx$$ Then $$\int_{a(t)}^{b(t)} g(x,t)\,dx=F(a(t),b(t),t)$$ which you can differentiate using the chain rule. It is easy enough to compute the partial derivatives with respect to $u$ and $v$ using the fundamental theorem of calculus.