Spivak - introducing the derivative to the integral Spivak writes that if a function $f$ is integrable on $[a,b]$, we can define a new function $F:[a,b] \to \Bbb{R}$ by the rule
\begin{align}
F(x) := \int_a^x f \equiv \int_a^x f(t) \, dt
\end{align}
(where I use "$\equiv$" to mean "same thing expressed in different notation").
He writes that this depends on a previous theorem, one which essentially stated that the integral of a function $f$ on $[a,b]$ could be broken up into two separate intervals, and vice versa, provided $f$ is integrable on $[a,b]$. 
The first part -- about "introducing" the derivative to the integral -- stumped me, and I haven't been able to find any insightful connection between the theorem and the equality. Could someone please provide some guidance or clarify how or why the derivative appeared?
On a side note, I suspect this newly defined function may also be related to the theorem immediately preceding it (a picture of the whole page has been pasted below), though beyond the "suppose now," I don't see how.

 A: Based on your latest comment, here's my answer: The presence of $dt$ in the integral is not meant to suggest any sort of derivative/differentiation. At the level you are studying, you should just take the notation $\int_a^x f(t)\, dt$ as just that - a piece of notation. This notation has been around for hundreds of years, and at times it is very handy, which is why we keep it around.
(This is why in my edit, I used the symbol "$\equiv$" and explicitly wrote that it means the same thing as $\int_a^x f$, just written differently)
In this book, Spivak more often uses the notation $\int_a^b f$ instead, and as he explains in the beginning of the chapter, this is equal to a certain real number which you calculate by taking the sup/inf of lower/upper sums. I suggest you CAREFULLY read through pages 261 and 262 of his book again, where he explains how to interpret the notation (I think his choice of words in explaining things is very clear, so pay close attention to it :).
Anyway, the purpose of the remark which Spivak made is to explain why the new function being defined, $F$, is actually well-defined (i.e why it makes sense from a logical perspective). What are we doing with the new function $F$? Well, we start by assuming $f$ is integrable on $[a,b]$. Then, we pick any $x \in [a,b]$, and we wish to define
\begin{align}
F(x) = \int_a^x f.
\end{align}
However, before we make such a definition, we need to ask ourselves: "does this make sense?" Look at what is on the RHS, we have the integral $\int_a^x f$, so we need to verify that this number actually exists! This is where Theorem 4 comes in: 

Theorem $4 \implies$ for any $x \in [a,b]$, $f$ is integrable on $[a,x]$ (i.e the real number $\int_a^x f$ exists).

It is because of this highlighted implication that we can define the function $F$ on $[a,b]$, and everything is logical (all the things actually exist).
A: I think you can think of $dt$ as more than just a "piece of notation".
First of all, think about integration as infinite summation.  An integration is merely the sum of an infinite number of infinitely small values.  
Areas are found by breaking up a shape into infinitely small boxes, and then adding them up.  So, if you have the function $f(t)$, and you want to know the infinitely small area at $t = t_0$, then the area itself is width ($dt$ - an infinitely-small width) times the height of the function ($f(t)$).  Put together, this is written $f(t)\,dt$. 
If we want to add up all the areas, we sum them.  But, since there are an infinite number of them, we use an integral.  Therefore, the area under the curve is the sum of all of the tiny areas, or $$\int f(t)\,dt$$
So, perhaps a better way of writing this would be to say, "if $f$ is integrable on $[a,b]$, then if we want to define a function for giving us the area under the curve of $f(t)$ from $a$ to some other value, we can see that this would be done by finding the infinitely small area at each point along the interval, and then summing them all up.  The infinitely small area at each point will be given by $f(t)\,dt$, and they can be summed from $a$ to $x$ using $$\int_a^x f(t)\,dt$$  Therefore, we can define $F(x)$ to be equivalent to this function."
A summary of looking at integrals this way can be found in Simplifying and Refactoring Introductory Calculus 
