In the last section, we talked about
the derivative of a function at a point.
Now we're gonna talk about the derivative
of a function as a function in
its own right.
Before we do that, let's review what
we learned about a derivative
at a point.
Let's suppose we've got a function,
y = f(x),
and we often like to draw the graph.
y = f(x), and we want to say, "What do 
we mean by the derivative of f(x)
at this point?"
Or our basic idea, was that it was the 
instantaneous rate of change of
f(x) at x = a.
As we increase x, how fast is 
f(x) changing?
We saw that that was the same thing
as the slope of a tangent line.
If we draw the line that's tangent to
the curve at this point and take its slope
in this case it looks like 
it's around (-1),
that's going to be the
instantaneous rate of change.
It also gives us a conversion factor.
If you increase x a little bit, how much
are you going to change f?
Roughly, you're gonna change it
by the slope * the run.
The rise is the slope * the run.
As long as you stay close to a,
you're going to stay close to the
tangent line.
That's a pretty good approximation.
It was a rate of change.
It was a slope.
It was a conversion factor.
It's all three things.
Now we wanna look at a function.
This is how I think of a function.
A function is a machine that takes
numbers and spits out numbers.
From this case, it might take the number 1
and spit out f'(1).
It might take the number 2 
and spit out f'(2).
It might take a number, a, 
and spit out f '(a).
We call this the f' function.
The idea is that the value 
of f' at a particular point
is the derivative of f(x)
at that point.
Usually when you have a function,
you wanna have a formula for
that function so you can actually
go out and compute it.
We know that the derivative of f
at a particular point, a,
is the limit as h goes to 0,
[ f(a+h) - f(a) ] / h.
There's the formula for 
our f'(x).
f'(x) is the same thing only with
x playing the role of h because
this is sort of the general form.
If you wanna do this and then 
plug in x = a, you get this.
And that's the derivative at a.
Let's work an example.
The example is a pretty simple function.
It's the function f(x)=x^2.
If we wanted to figure out what was the
derivative at 1, we know how to do that.
We take f of [(1+h)^2] - f(1),
that's one,
divide by h, and take a limit.
We do a little algebra,
we expand that out.
We discover that the 1's cancel.
We get ((h^2)+2h)/h.
That's (h+2), take a limit as h goes to 0
and you get 2. Wonderful.
f'(1) is 2.
If you wanted to know what f'(5) was,
you could do the same kind of calculation.
{[(5+h)^2] - (5^2)} / h.
You expand it out.
The 25's cancel.
You divide by h.
You get (10+h).
Take the limit as h goes to 0,
you get 10.
If I'm going to fast for you,
pause the thing, look at the screen,
work it out yourself,
and then hit play again.
We could do this at an arbitrary point.
It's really the same calculation 
we did at 1 or at 5.
{ [(a+h)^2] - (a^2) }, well let's see.
a + (h^2) is 
(a^2)+(h^2)+(2ah), you subtract off (a^2)
so you have (h^2) / h, that's h,
(2ah) / h, that's 2a.
The limit of (2a+h) is 2a.
Instead of calling it a, 
we could've called it x.
So f'(x), the derivative of x^2,
is given by this calculation.
We take { [f(x+h)^2] - f(x) } / h.
Expand it out.
The (x^2)'s cancel.
[(2xh)+(h^2)] / h is 
(2x+h),
we take the limit of that as
h goes to 0,
and we get 2x.
Now we can draw a picture
of what we've just done.
We started with a function,
y = (x^2).
And we derived a new function out of that.
And the new function was 
f'(x) = 2x.
We should give this a name 
and call it g(x).
So the function g(x) was derived 
from f(x) and that's why we call it
the derivative function.
If you look at any point in the curve
like over here, you'd say
what's the derivative of y=(x^2)?
Oh yeah. It looks like the slope 
is about -
Say this is 0.1, the slope is about 2.
Sure enough, the value of the 2x function
is the slope of the x^2 function.
Over here, the slope is 0.
Over here, the value is 0.
Over here, the slope is negative.
It's going downhill.
Over here, the value is negative.
The whole idea of the derivative function
is the value of the derivative function
at a point gives you the derivative
of the original function at that point.
We'll work out a bunch more
examples in the next video.
