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## Derivative as a function

# Connecting f and f' graphically

## Video transcript

So what we have plotted here is
some function-- let's call it f of x-- and its
derivative, f prime of x. And what we need
to figure out is which one is f of x and
which one is f prime of x? So let's take a stab at it. Let's think about what
would be the situation if the green
function were f of x. So let's see if this works out. If the green function were f
of x, does the orange function here, or the yellow function,
could that be f prime of x? So let's think about
what's happening to this green function
at different points. So this green function
right over here, right at this point if
we start at the left, has a positive slope. If this orange
function were f prime of x, if it were the derivative
of the green function, then it would have
to be positive because the green
function's slope is positive at that point. But we see that
it's not positive. So it's pretty clear that the
green function cannot be f of x, and the yellow function
cannot be its derivative, because if this
was its derivative, it would be positive here. So that quickly, we found out
that that can't be the case. But let's see if it could
work out the other way. So it's starting to feel--
just ruling that situation out-- that maybe that this is
f of x and the green function is f prime of x. So let's see if this
holds up to scrutiny. So what we have when we
start off at the left, f of x, or what we
think is f of x, has a reasonably positive slope. Is that consistent? Well, yeah, sure. Our green function is positive. In fact, at the
point, it's telling us that the slope of the tangent
line is around 2 and 1/2. And it actually does look like
the slope of the tangent line is exactly 2 and 1/2 of
this function right here. Actually, let me
erase this, just so we don't look
like we're trying to take the slope of the
tangent line of the derivative. So it looks just like that. So we see the slope of the
tangent line right over here looks like about 2 and 1/2,
and the value of this function up here looks like
it's about 2 and a 1/2. So, so far this
green function looks like a pretty good
candidate for the derivative of this yellow function. But let's keep going here. So let's think
about what happens as we move to the right. So here, let's see. It looks like the slope
of this yellow function-- let me just use a color we
can see-- it keeps going up. It keeps going up,
keeps going up. And then at some point, it
reaches some maximum slope, and then it starts
to go down again. The slope starts to go
down again, all the way to the slope going all the
way down to 0 right over here. Well, does this green
function describe that? Well, let's see. The slope is positive
and increasing up to this point, which seemed
pretty consistent with what we just experienced. Then the slope stays
positive, but it's positive and decreasing. And that's what we saw here. The slope is positive and
decreasing all the way to the slope getting to 0
at this maximum point here. And we see, indeed, that
on this green function, the green function hits 0. So it seems like it's doing
a pretty good job of plotting the slope of the tangent
line of the orange function. And then our slope becomes
more and more negative. And then it hits some
point, some minimum point right over here. The slope hits some minimum
point right over here. And then it becomes
less and less negative. Let's see how well
I can draw it. The slope becomes
less and less negative until it hits a 0 slope again. And then it starts
becoming positive until it hits some
maximum slope. But then it stays positive,
but it becomes less positive. It becomes less
and less positive. So it looks pretty clear that
the orange function is f of x, and the green function
is f prime of x.