Welcome to the presentation on derivatives.
I think you're going to find that this is when math starts
to become a lot more fun than it was just a few topics ago.
Well let's get started with our derivatives.
I know it sounds very complicated.
Well, in general, if I have a straight line-- let me see if I
can draw a straight line properly-- if I had a straight
line-- that's my coordinate axes, which aren't straight--
this is a straight line.
But when I have a straight line like that, and I ask you to
find the slope-- I think you already know how to do this--
it's just the change in y divided by the change in x.
If I wanted to find the slope-- really I mean the slope is the
same, because it is a straight line, the slope is the same
across the whole line, but if I want to find the slope at any
point in this line, what I would do is I would pick a
point x-- say I'd pick this point.
We'd pick a different color-- I'd take this point, I'd pick
this point-- it's pretty arbitrary, I could pick any two
points, and I would figure out what the change in y is-- this
is the change in y, delta y, that's just another way of
saying change in y-- and this is the change in x.
delta x.
And we figured out that the slope is defined really as
change in y divided by change in x.
And another way of saying that is delta-- it's that triangle--
delta y divided by delta x.
Very straightforward.
Now what happens, though, if we're not dealing
with a straight line?
Let me see if I have space to draw that.
Another coordinate axes.
Still pretty messy, but I think you'll get the point.
Now let's say, instead of just a regular line like this, this
follows the standard y equals mx plus b.
Let's just say I had the curve y equals x squared.
Let me draw it in a different color.
So y equals x squared looks something like this.
It's a curve, you're probably pretty familiar with it by now.
And what I'm going to ask you is, what is the
slope of this curve?
And think about that.
What does it mean to take the slope of a curve now?
Well, in this line, the slope was the same throughout
the whole line.
But if you look at this curve, doesn't the
slope change, right?
Here it's almost flat, and it gets steeper steeper steeper
steeper steeper until gets pretty steep.
And if you go really far out, it gets extremely steep.
So you're probably saying, well, how do you figure out
the slope of a curve whose slope keeps changing?
Well there is no slope for the entire curve.
For a line, there is a slope for the entire line, because
the slope never changes.
But what we could try to do is figure out what the
slope is at a given point.
And the slope at a given point would be the same as the
slope of a tangent line.
For example-- let me pick a green-- the slope at this point
right here would be the same as the slope of this line.
Right?
Because this line is tangent to it.
So it just touches that curve, and at that exact point, they
would have-- this blue curve, y equals x squared, would have
the same slope as this green line.
But if we go to a point back here, even though this is a
really badly drawn graph, the slope would be
something like this.
The tangent slope.
The slope would be a negative slope, and here it's a positive
slope, but if we took a point here, the slope would
be even more positive.
So how are we going to figure this out?
How are we going to figure out what the slope is at any point
along the curve y equals x squared?
That's where the derivative comes into use, and now for the
first time you'll actually see why a limit is actually
a useful concept.
So let me try to redraw the curve.
OK, I'll draw my axes, that's the y-axis-- I'll just do it in
the first quadrant-- and this is-- I really have to find a
better tool to do my-- this is x coordinate, and then let
me draw my curve in yellow.
So y equals x squared looks something like this.
I'm really concentrating to draw this at
least decently good.
OK.
So let's say we want to find the slope at this point.
Let's call this point a.
At this point, x equals a.
And of course this is f of a.
So what we could try to do is, we could try to find
the slope of a secant line.
A line between-- we take another point, say, somewhat
close, to this point on the graph, let's say here, and if
we could figure out the slope of this line, it would be a
bit of an approximation of the slope of the curve
exactly at this point.
So let me draw that secant line.
Something like that.
Secant line looks something like that.
And let's say that this point right here is a plus h, where
this distance is just h, this is a plus h, we're just going
h away from a, and then this point right here
is f of a plus h.
My pen is malfunctioning.
So this would be an approximation for what the
slope is at this point.
And the closer that h gets, the closer this point gets to
this point, the better our approximation is going to be,
all the way to the point that if we could actually get the
slope where h equals 0, that would actually be the slope,
the instantaneous slope, at that point in the curve.
But how can we figure out what the slope is when h equals 0?
So right now, we're saying that the slope between these two
points, it would be the change in y, so what's
the change in y?
It's this, so that this point right here is-- the x
coordinate is-- my thing just keeps messing up-- the x
coordinate is a plus h, and the y coordinate is f of a plus h.
And this point right here, the coordinate is a and f of a.
So if we just use the standard slope formula, like before, we
would say change in y over change in x.
Well, what's the change in y?
It's f of a plus h-- this y coordinate minus this y
coordinate-- minus f of a over the change in x.
Well that change in x is this x coordinate, a plus h, minus
this x coordinate, minus a.
And of course this a and this a cancel out.
So it's f of a plus h, minus f of a, all over h.
This is just the slope of this secant line.
And if we want to get the slope of the tangent line, we would
just have to find what happens as h gets smaller and
smaller and smaller.
And I think you know where I'm going.
Really, we just want to, if we want to find the slope of this
tangent line, we just have to find the limit of this
value as h approaches 0.
And then, as h approaches 0, this secant line is going to
get closer and closer to the slope of the tangent line.
And then we'll know the exact slope at the instantaneous
point along the curve.
And actually, it turns out that this is the definition
of the derivative.
And the derivative is nothing more than the slope of a
curve at an exact point.
And this is super useful, because for the first time,
everything we've talked about to this point is
the slope of a line.
But now we can take any continuous curve, or most
continuous curves, and find the slope of that curve
at an exact point.
So now that I've given you the definition of what a derivative
is, and maybe hopefully a little bit of intuition, in the
next presentation I'm going to use this definition to actually
apply it to some functions, like x squared and others, and
give you some more problems.
I'll see you in the next presentation