TITLE: How do I differentiate this integral?
QUESTION [5 upvotes]: That is:
$$\left(\int_{a(x)}^{b(x)}\!f(x,t)\,dt\right)'$$
I don't know how to differentiate a integral if functions of $x$ are at its limits.
Can you guys show me how to do this?

REPLY [3 votes]: There is an intermediate function of three variables involved, namely
$$F(u,v,w):=\int_u^v f(w,t)\ dt\ .$$
One has
$$F_u(u,v,w)=-f(w,u)\ ,\quad F_v(u,v,w)=f(w,v)\ ,\quad F_w(u,v,w)=\int_u^v f_w(w,t)\ dt\ ,$$
where the last formula is Leibniz' Rule "without extras".
When the variables $u$, $v$, $w$ become functions of $x$:
$$u(x):=a(x)\ ,\quad v(x):=b(x)\ ,\quad w(x):=x\ ,$$
then the composition with $F$ defines a function $\phi(x):=F\bigl(a(x),b(x),x\bigr)$. In order to compute the derivative $\phi'$ we have to use the chain rule and obtain the formulas given in Nana's answer.