Definition of the Derivative

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.