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## Derivative as a function

# The graphical relationship between a function & its derivative (part 1)

## Video transcript

So I've got this crazy
discontinuous function here, which we'll call f of x. And my goal is to try to draw
its derivative right over here. So what I'm going to
need to think about is the slope of
the tangent line, or the slope at each
point in this curve, and then try my best
to draw that slope. So let's try to tackle it. So right over here at this
point, the slope is positive. And actually, it's
a good bit positive. And then as we get larger
and larger x's, the slope is still positive, but it's less
positive-- and all the way up to this point right over
here, where it becomes 0. So let's see how I could
draw that over here. So over here we
know that the slope must be equal to
0-- right over here. Remember over here,
I'm going to try to draw y is equal
to f prime of x. And I'm going to
assume that this is some type of a parabola. And you'll learn shortly why
I had to make that assumption. But let's say that,
so let's see, here the slope is quite positive. So let's say the slope
is right over here. And then it gets less and
less and less positive. And I'll assume it does
it in a linear fashion. That's why I had to assume that
it's some type of a parabola. So it gets less and
less and less positive. Notice here, for example,
the slope is still positive. And so when you look
at the derivative, the slope is still
a positive value. But as we get larger and
larger x's up to this point, the slope is getting
less and less positive, all the way to 0. And then the slope is getting
more and more negative. And at this point, it seems like
the slope is just as negative as it was positive there. So at this point
right over here, the slope is just
as negative as it was positive right over there. So it seems like this
would be a reasonable view of the slope of the tangent
line over this interval. Now let's think about
as we get to this point. Here the slope seems constant. Our slope is a constant
positive value. So once again, our slope here
is a constant positive line. Let me be careful here
because at this point, our slope won't really be
defined, because our slope, you could draw
multiple tangent lines at this little pointy point. So let me just draw a
circle right over there. But then as we get
right over here, the slope seems to be positive. So let's draw that. The slope seems to be
positive, although it's not as positive as it was there. So the slope looks like it is--
I'm just trying to eyeball it-- so the slope is a constant
positive this entire time. We have a line with a
constant positive slope. So it might look
something like this. And let me make it clear what
interval I am talking about. I want these things to match up. So let me do my best. So this matches up to that. This matches up over here. And we just said we have
a constant positive slope. So let's say it looks something
like that over this interval. And then we look at this
point right over here. So right at this point, our
slope is going to be undefined. There's no way that you
could find the slope over-- or this point of discontinuity. But then when we
go over here, even though the value of our
function has gone down, we still have a
constant positive slope. In fact, the slope
of this line looks identical to the
slope of this line. Let me do that in
a different color. The slope of this
line looks identical. So we're going to continue
at that same slope. It was undefined at
that point, but we're going to continue
at that same slope. And once again,
it's undefined here at this point of discontinuity. So the slope will look
something like that. And then we go up here. The value of the
function goes up, but now the function is flat. So the slope over
that interval is 0. The slope over this interval,
right over here, is 0. So we could say--
let me make it clear what interval I'm
talking about-- the slope over this interval is 0. And then finally, in
this last section-- let me do this in orange--
the slope becomes negative. But it's a constant negative. And it seems actually a
little bit more negative than these were positive. So I would draw it
right over there. So it's a weird
looking function. But the whole
point of this video is to give you an intuition
for thinking about what the slope of this function
might look like at any point. And by doing so,
we have essentially drawn the derivative
over that interval.