Dive Deep into Second Fundamental Theorem of Calculus with Comprehensive Examples and Expert Assistance

Explain the meaning of derived unit with the help of one example.

Which of the following is TRUE ?
A
The fundamental period of $\mid <!--\; \mid \; -->\sin <!--\; \; -->x\mid <!--\; \mid \; -->+\mid <!--\; \mid \; -->\cos <!--\; \; -->x\mid <!--\; \mid \; -->$
B
The fundamental period of $\mid <!--\; \mid \; -->\sin <!--\; \; -->x\times <!--\; \times \; -->\cos <!--\; \; -->x\mid <!--\; \mid \; -->$
C
The fundamental period of $\mid <!--\; \mid \; -->\sin <!--\; \; -->x\mid <!--\; \mid \; -->-<!--\; -\; -->\mid <!--\; \mid \; -->\cos <!--\; \; -->x\mid <!--\; \mid \; -->$
D
The fundamental period of $\mid <!--\; \mid \; -->\sin <!--\; \; -->x\mid <!--\; \mid \; -->$

Second Fundamental theorem of calculus
Theorem 3.20 (Second Foundamental Theorem of Calculus)
Let f be a continuous function on [a,b] and F any function on [a,b], differentiable on (a,b), continuous on [a,b] such that $F^{\prime }(x)=f(x)$ for all $x\in <!--\; \in \; -->(a,b)$.
Then
${\int <!--\; \int \; -->}_a^{b}f(x)\mathrm{d}x=F(b)-<!--\; -\; -->F(a)$
I need to use the second Fundamental theorem of calculus to work out:
${\int <!--\; \int \; -->}_0^{\frac{\mathrm{\pi <!--\; \pi \; -->}}{8}}\tan <!--\; \; -->(2x)\mathrm{d}x$
Firstly it is clear that tan(2x) is continuous on $[0,\frac{\mathrm{\pi <!--\; \pi \; -->}}{8}]$
Now $F(x)=-<!--\; -\; -->\frac{1}{2}\ln <!--\; \; -->|\cos <!--\; \; -->(2x)|=-<!--\; -\; -->\frac{1}{2}\ln <!--\; \; -->\cos <!--\; \; -->(2x)$
where$F^{\prime }(x)=f(x)$
To show that F(x) is differentiable $\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->(0,1)$ is it enough to say that as f(x) is continuous on (0,1) the derivative exists?

Difference between first and second fundamental theorem of calculus
In first fundamental theorem of calculus,it states if $A(x)={\int <!--\; \int \; -->}_a^{x}f(t)dt$ then $A^{\prime }(x)=f(x)$.But in second they say ${\int <!--\; \int \; -->}_a^{b}f(t)dt=F(b)-<!--\; -\; -->F(a)$,But if we put x=b in the first one we get A(b).Then what is the difference between these two and how do we prove A(b)=F(b)−F(a)?

The notation of the second fundamental theorem of Calculus
I am self studying calculus, and just finished the lesson on the second fundamental theorem of calculus.
the way the theorem is described is:
$\frac{d}{dx}({\int <!--\; \int \; -->}_a^{x}f(t)dt)=f(x)$
and it was told that the meaning is that the derivative of an integral of a function is the function itself.
I don't get how you can get that from this. the expression that I would think suggests this is:
$\frac{d}{dx}(\int <!--\; \int \; -->f(x)dt)=f(x)$
so the derivative of an indefinite integral (as oppose to integrating over a range) of a function is the function itself.
another interpretation of the FToC2 I read here, is that it means that the derivative of the functions that gives the area under the curve of a different function is the different function. this is also something I don't understand how the FToC2 suggests of?
to me, it seems like what this means:
$\frac{d}{dx}({\int <!--\; \int \; -->}_a^{x}f(t)dt)=f(x)$
is how a very small change in x affects that area under f(t) between a (a constant) and x. how do I get from that to the right interpretation?

Second Fundamental Theorem of Calculus...
Let f have a continuous second derivative. Prove that
$f(x)=f(a)+(x-<!--\; -\; -->a)f^{\prime }(a)+{\int <!--\; \int \; -->}_a^x(x-<!--\; -\; -->t)f^{″}(t)dt.$
Here is my attempt at the problem.
Since f has a continuous second derivative, then the first derivative is also continuous. Therefore, by the first fundamental theorem of calculus, we have that
$f(x)=f(a)+{\int <!--\; \int \; -->}_a^x{f}^{\prime }(t)dt.$
Expanding out the right-hand side of the above using integration by parts, we see that
$f(x)=f(a)+f^{\prime }(t)t-<!--\; -\; -->{\int <!--\; \int \; -->}_a^{x}t{f}^{″}(t)dt.$
This is where I am confused.

fundamental theorem of calculus and second derivative
Using the fundamental theorem of calculus, find the second derivative of
${\int <!--\; \int \; -->}_{\sqrt{(}x)}^{x}x-<!--\; -\; -->e^{t}dt$
I've looked up the theorem on wikipedia but I can't really see what I'm meant to do.