Let $I$ and $J$ be intervals on $\mathbb{R}$, and $f: I \times J \rightarrow \mathbb{R}$ a continuous function that has a partial derivative with respect to the first variable, and this partial derivative is also continuous on $I \times J$. Let also $a,b : I \rightarrow J$ derivable functions. Show that $F: I \rightarrow \mathbb{R}$ defined as: $$F(x)= \int^{b(x)}_{a(x)}f(x,t)dt $$ is derivable.
So my first approach was to study the behaviour of $F(x+h)$: $$F(x+h) = \int^{b(x+h)}_{a(x+h)}f(x+h,t)dt $$ The issues I have here is that I don't know how $a,b$ behave. Any hint will be appreciated.
Fix a number $h\in J$. Then$$F(x)=\int_h^{b(x)}f(x,t)\,\mathrm dt-\int_h^{a(x)}f(x,t)\,\mathrm dt.$$So, if$$g(x)=\int_h^xf(x,t)\,\mathrm dt,$$then $F(x)=g\bigl(b(x)\bigr)-g\bigl(a(x)\bigr)$ and this expresses $F$ in terms of $g$, $a$ and $b$ using only composition of functions and algebrauc operations. SInce $a$, $b$, and $g$ are differentiable, $F$ is differentiable too.