Uploaded by MSTEOffice on 14.04.2011

Transcript:

Now the derivative is one of the two most fundamental concepts in calculus.

The derivative is one, and the other one is the integral.

And the derivative is the basis of something called the differential calculus and you may

wonder why it's by the differential calculus

instead of the derivative calculus

and that's for historical reasons because when the calculus was first invented

people thought in terms of something called differentials uh... which uh... later went

out of style. Although, they've come back more recently.

The purpose of the derivative

is to measure the rate of change of things. So, if you're interested in change, and you wanna

know how things are changing,

uh... it's the derivative that uh... that you want.

Now,

one of the things the derivative does,

and this was the original historical reasons

for why people were interested in it, in the

the seventeenth-century, is because it tells

It also enables you

the construct tangent lines to curves.

That is not what its...

that is not the cause of its being important today,

but i'm going to use that

idea of to derivative tells you the slope of the tangent line to a curve,

because it makes the whole discussion

very visual, and you can actually see

the derivative whereas if you do other kinds of applications,

uh... you have to uh... there's not a clear visual picture of it.

So,

what I'm gonna do is I'm gonna make a general definition,

and then and then they give two examples. There are two ways to proceed always in mathematics

You can

You can either give examples first, and then

from those, extract the definition, or you can give a definition first, and then give some

examples.

In this case, I'll do the definition first.

And so, we began.

uh...

by drawing a picture.

And then, I'm gonna call that function

y = f(x),

and I'm interested in

knowing

the slope of the tangent line

at some point x.

And I will draw in that tangent line so you can see it.

And I wanna know what the slope of that thing is. So, I'll write m_tangent.

And the idea behind all this is

that

we're gonna look at secant lines, and then, we'll go from there. So, I'm gonna move here

from x to

I'm gonna take some quantity

h,

and I'm gonna... well, I'm gonna move over

to x + h.

We'll make that over here.

And...

by the way,

I'm gonna call this point P

and that of course is the point

(x,f(x)).

(x,f(x)).

And other here

I have the point Q,

which is what?

Well, its x coordinate is x+h,

and its y coordinate is

f(x+h)

f(x+h).

So, that's what this point is.

And now,

in green,

I'm gonna draw the secant line that connects P and Q.

and that secant line has a slope,

and what would that slope be? Well,

slopes are usually denoted by m.

So, I'll write

m_PQ,

and what is that?

Well, the slope of a line is the difference in the y coordinates over the difference

in the x coordinates.

So, what is that? That's

f(x+h) - f(x).

f(x+h) - f(x).

That's the difference in the y's,

And what is the difference in the x's? I can

I'm not gonna right write (x+h) - x, 'cause I can do that in my head.

And the answer is h.

So, that's the slope of the secant lines.

And now, what I'm gonna imagine is

I'm gonna imagine

taking this point Q

Think of this

as a wire...

the function itself.

Think of this as a little bead on that wire

like they have in pool rooms

where you keep score, and I'm gonna take this little bead

and I'm gonna move it

towards P.

So, the next time I look at it, maybe it will be here,

and now, it will look like that.

And the next time I look at it, it will be here,

and it will look like this.

So, here is a whole bunch

of secants.

In fact, if you're so inclined, mathematicians would call that a pencil of secant lines.

And hopefully, and I probably should extend them because they go down here.

Now,

Q gets closer and closer to P,

I hope that you will, that I can persuade you that

those green lines become more and more like the red line.

So that in the limit,

they actually

become the redline. Let me write that observation down.

The limit

as Q approaches P

of the slopes

of PQ

or in other words

looking at the algebra of it...

What do I have to do here in order for that Q to move towards P?

I have to do what?

I have to let this h get small.

And those limits then in fact give you the slope of the tangent line at P.

And this slope

algebraically is known as the derivative

of y = f(x) at the point...

when x is at x.

And so, and that's always denoted by

f'(x)

f'(x)

And that is then the derivative

of

f(x).

And so, that's what it is.