Problem 3-33.
If $f:[a,b]\times [c,d]\to\mathbb{R}$ is continuous and $D_2f$ is continuous, define $F(x,y)=\int_a^x f(t,y)dt$.
(a) Find $D_1F$ and $D_2F$.
(b) If $G(x)=\int_a^{g(x)} f(t,x)dt$, find $G'(x)$.
I solved (a) as follows:
$D_1F(x,y)=f(x,y)$ by a famous theorem in one variable calculus.
$D_2F(x,y)=\int_a^x D_2f(t,y)dt$ by Problem 3-32.
But I could not solve (b).
At first, what is $g$?
I guess $g$ is a function from $[c,d]$ to $[a,b]$ and $g$ is differentiable on $[c,d]$.
$G(x)=F(g(x),x)$.
$G'(x)=D_1F(g(x),x)\cdot g'(x)+D_2F(g(x),x)=f(g(x),x)\cdot g'(x)+\int_a^{g(x)} D_2f(t,x)dt$.
I cannot understand the author's intention for this problem at all.