Welcome, let's quickly recap the main ideas from Section 1.4 in Active Calculus
on the derivative function.
So previously in the course we've defined the derivative of a function, f,
at a specific point, a, to be f prime of a, and that's equal to the limit
as h approaches zero, of f of a plus h minus f of a all divided by h.
In this section, we're not really doing anything new to the formula.
We're simply observing that if we let a be a number whose value we don't specify
until later, rather than a specific number that we specify right now, then we can
go through the same limit process to arrive at a formula for f prime of a.
We can then specify the value of a, and
very quickly calculate a lot of derivatives all at once in
different points without having to retake a limit every time.
For example, we saw it in the book that if f of x is four x minus x squared,
and f prime of a is four minus 2a, no matter what the value of a is.
So again, this makes it very easy to compute derivative values because instead
of having to calculate a limit every time we needed a value of f prime of a.
We just calculate the limit once and for all using an unspecified value of a, and
then we get a formula for f prime of a, and
use that result in formula to calculate the derivative at specific points.
The trade-off here is that the algebra involved in coming up with f prime of a in
this way can be a little more complicated.
But if you're willing to pay that price, then the payoff is pretty good.
So since we can calculate f prime of a for
any value of a, that makes the derivative of f a brand new function unto itself.
We call this the derivative function, f prime of x.
And we're now using x to make it clear that this derivative is a function with
a variable input and not just a fixed point like specifically one or
specifically negative five or so on.
And we're going to define the derivative function f prime of x, no surprise to be
the limit as h goes to zero, of f of x plus h, minus f of x divided by h.
Provided that the limit exists.
That's the one main concept of this section.
There are two applications of this concept in this section, they just have to do with
treating f prime of x, the derivative function like any other function.
First, we need to be able to find a formula for
f prime of x if we're given a formula for f of x.
And second, if we're given a graph of f of x,
we'd like to be able to sketch the graph of f prime of x.
So both of those tasks will have examples in the videos that are coming up right now
in addition to the ones in the textbook,
