Fundamental theorem of calculus for a function depending on the same variable of integration bound.

Question

Let $r, a,b\in\mathbb{R}$ and $f\in L_1 ([a,b])$. Define the function $$F : [a, b] \to\mathbb{R}$$ as $$F (x) = r + \int_{r}^{x} f(t)\,dt$$ (so that $F (a) = r$). The Fundamental Theorem of Calculus pt. I states $$F'(x) = f(x).$$

My question is: If a function $g$ is defined in two variables with domain $[a, b]\times [a, b]$ and $$G(x):= r + \int_{r}^{x} g(x,t)f(t)\,dt,$$ then what is $G'$?

Answer 1

It is $$ G'(x) = g(x,x)f(x) + \int_r^x \frac{\partial g}{\partial x}(x,t) f(t)\,dt. $$ This follows from the Leibniz integral rule.