We're going to start by talking about the idea of a derivative.

And in particular the deriviatve as the limit of the slope of a line joining two

points on a curve. Now the, let's starat by just drawing any

old curve f of x versus x. Now let's focus our attention on a

certain point, we'll call that x0, and the function has a value f of x0 at that

point. Now what we're trying to do is find the

instantaneous rate of change of the of this curve, and what that's equivalent to

is the slope of the tangent line at that point.

Now to find an expression for that, we're going to start by considering another

point on the curve, delta x. Further along the x axis than x zero.

And the function has a value at that point.

And what I want to do is concstruct a straight line that joins those two points

and I want to write down an expression for the slope of that straight line.

So, The slope of this line is just the, the rise over the run.

So it's the change in y value divided by the change in x value.

So the change in the y value is just f, at x0 plus delta x, minus f of x0, so

that's the delta y. And then the Delta x is x0 plus Delta x

minus x0, down here. Now if I cancel the x0 factors I'm left

with this. It's just the change in f of x divided by

Delta x. Now what I want to do is take the limit

as delta goes to 0. So I'm going to move this point down the

curve, and as I do that the line joining the two points is going to approach this

tangent line. And the tangent line then will tell you

How much the f of x is changing when I have a very small change in x.

That's the instantaneous rate of change of this function evaluated at the point x

to 0. So if I take the limit as delta x goes to

0 of this expression up here. That's the thing that we're going to

define as the derivative. Of the function f of x, evaluated at x

zero. And the shorthand notation for that is f

prime, so f prime means the derivative of the function f.

And that f prime is evaluated at x zero in this case.