12 The Derivative
Of course, one of the main reasons for looking at limits in calculus is to use them to define the derivative of a function. The derivative is covered in sections 3.1 through 3.3 of the textbook. There is little here, really, that was not already covered in calculus, except that the level of rigor is higher.
Definition: Let $f$ be a real-valued function of a real variable, and let $a\in \R$. Assume that $f$ is defined on some open interval containing $a$. Then the derivative of $f$ at $a$ is defined to be $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$ provided the limit exists. If the limit exists, then $f$ is said to be differentiable at $a$. The function $f'$ is called the derivative of $f$, and its domain is the set of all points in the domain of $f$ at which $f$ is differentiable.
It is then possible to prove that if $f$ is differentiable at $a$, then $f$ must be continuous at $a$. It is also possible to prove the usual differentiation rules for sums, differences, products, and quotients, as well as the chain rule for the derivative of a composition $f\circ g$.
The Mean Value Theorem (MVT) is a geometrically obvious fact that turns out to be important in the proofs of a number of future calculus theorems, such as L'Hôpital's rule. There is also a Generalized Mean Value Theorem that applies to two functions.
Theorem (Mean Value Theorem): Suppose the function $f$ is continuous on the closed, bounded interval $[a,b]$ and is differentiable on $(a,b)$. Then there is a $c\in(a,b)$ such that $$f'(c)=\frac{f(b)-f(a)}{b-a}$$
Theorem (Generalized Mean Value Theorem): Suppose the functions $f$ and $g$ are continuous on the closed, bounded interval $[a,b]$ and are differentiable on $(a,b)$. Then there is a $c\in(a,b)$ such that $$(g(b)-g(a))f'(c)=(f(b)-f(a))g'(c)$$
It is the Generalized Mean Value Theorem that is used to prove L'Hôpital's rule. Note that if it happens that $g'(c)\ne 0$ and $g(b)\ne g(a)$, then the conclusion can be written $$\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$ In particular, applying the generalized theorem to the functions $f(x)$ and $g(x)=x$ yields the basic Mean Value Theorem for $f(x)$.