Fundamental theorem of calculus related problem Let
$$F(t) = \int ^t_0 f(s,t)\, ds$$
How can I find out $F'(t)$?
I guess we should apply chain rule but I can't succeed. Well, perhaps, separation of variables?

Let $$G(u,v)=\int_0^uf(s,v)\,ds$$ Note that if $u(t)=t$ and $v(t)=t$, then $F(t)=G(u,v)$. Now $$
\begin{align}
\frac{d}{dt}F(t)&=\frac{d}{dt}G(u,v)\\&=\frac{\partial G}{\partial u}\frac{du}{dt}+\frac{\partial G}{\partial v}\frac{dv}{dt}\\
&=f(u,v)\cdot1+\int_0^u\frac{\partial}{\partial v}f(s,v)\,ds\cdot1\\
&=f(t,t)+\int_0^t\left.\frac{\partial}{\partial v}f(s,v)\right|_{v=t}\,ds\\
\end{align}$$
