real analysis - Application of Fundamental Theorem of Calculus, splitting integrals

Looking for some help and explanation on some questions that require FTC.
I will be using the following versions of FTC. I have made an attempt but they do not correspond to the solutions I was given out so am looking for feedback on why my method is incorrect.

FTC 1

Let $f$ be regulated on the interval $[a,b]$ and $f$ is continuous on at $c \in (a,b)$. Then $F(x) = \int^x_a f$ is differentiable on $(a,b)$ with $F'(c) = f(c)$

FTC 2

Let $f: [a,b] \rightarrow \mathbb{R}$ be continuous and let $g: [a,b] \rightarrow \mathbb{R}$ be differentiable with $g'(x) = f(x) \forall x \in [a,b]$. Then $\int^b_a f = g(b) - g(a)$

Questions

1) Let $I = \int^b_a \frac{1}{log x}dx$
Find: $\frac{dI}{da}, \frac{dI}{db}$

So my method is to let $F$ be such that $F' = \frac{1}{log x}$, and then using FTC2, I get $I = F(b) - F(a)$. After differentiation I get $\frac{1}{\log a}, -\frac{1}{\log b}$ respectively.

Now the solutions suggest that I split the integral up into $\int^b_2, \int^2_a =\int^b_2 - \int^a_2$, and then differentiate here. Am I missing something about the conditions? Do I need to rearrange each equation into the indefinite integral?

2) Same as above except now $I = \int^{b(x)}_{a(x)} \frac{1}{\log x} dx$ I need to find $I'(x)$ $a(x)$ and $b(x)$ are functions with values in $[2, \infty)$

In the form of b(x), a(x) I can't seem to rearrange it into the indefinite integral form. From the solutions they still seem to recommend that I write
$I(x) = J(a(x)) - J(b(x))$ where $J(a) = \int^a_2 \frac{1}{log t} dt$ and then differentiating with the chain rule. I don't understand why this is necessary, as to me, an unrigorous solution that uses the same idea $\frac{d}{dx}(F(b(x)) - F(a(x))$ does the same thing, so why do we need to split the integral like this?

3) Can someone give me an example of how they would split $\int^{1+x^2}_{x^2}f(t) dt, x \in \mathbb{R}$

4) Where do I need to comment about differentiability / continuity? Do I need to prove that the functions I here are continuous before I use the theorem?