# Derivative intuition module

## Video transcript

This right here is our
derivative intuition module. And it was a module that was
contributed by Ben Eater, and it's a pretty interesting
module, one, it's graphical and all the rest. But what's really
cool about it is it really helps to conceptualize
what a derivative is all about. And all it really
is is the slope of the tangent
line at any point. So in this exercise
right over here, they've graphed f of x is
equal to 2x to the third power, so this is 2x to the third
power right over here. And what we need to do is move
these orange dots up and down, so these orange dots
right now show a line. So when I highlight
on that orange dot, you saw this line down
here became orange. And right now that
orange dot is 0 because the slope of this
line right over here is 0. But we see that that line right
now is definitely not tangent, or the slope at that point
in our curve 2x to the third, is definitely not 0. So what we need to do
is we need to move this, so it becomes pretty close, as
close as we can kind of eyeball it, to the slope of
the tangent line. And so that looks about-- right
about there looks about right. That looks pretty
close to the slope of the curve at that point. And you could even see
while I do it, up here, you see DDX of f of negative 2. So the derivative at f
is equal to negative 2. So that's just saying
that when the function is equal to negative 2, what
is the derivative there? That's the slope. So by putting it
all the way up here, I'm saying that the slope is
equal to 24 at that point. So that's a pretty good
approximation just eyeballing it like that. Now, let's do this
point right over here. So it looks like the slope
is decreasing a little bit, but it's still very positive. So the slope looks maybe it's
close to around 14 or 15. And then over here,
once again, our slope has decreased bit it's
still reasonably positive. The slope looks
like it's about 5, so the slope is a decreasing. Right over here
the slope does look like it's about 0
right over here. This looks like an
inflection point actually. So our slope is 0 right there. And you see the
derivative at f is 0 is 0, which just means
this is fancy notation to say that the slope
of the tangent line, or you could say the slope of
the curve at that point, is 0. It's flat there, and you
see it really is flat. And then you go
over here, and we want to find the
derivative at f of 1. And so once again, it looks like
the slope is now increasing. So this is interesting. Up here, the slope
was very positive, but it was decreasing
as we went here. Notice, it's getting
flatter and flatter. Then it goes to 0, and now the
slope starts increasing again. So at this point, we
want to make it tangent. And so now notice
the slope has gone up relative to
when we were at 1. So that looks like the
slope right over there. And then we do the
slope over there. And if we get
pretty close, it'll actually draw the curve
of the derivative. So there you go. And so what's neat
about here, we didn't find the derivative
at every point in the curve, we found it at just add
the special points that we were able to move
these orange dots for. But since we got the
orange dots close enough to the actual derivatives,
just by eyeballing it, just trying to find the--
trying to look at what the slope of this
tangent line is, it said, OK, you've done it
right, and it drew the whole slope
of the derivative. And what this entire derivative
is saying is at any point, this is the slope
of the tangent line. So even though we didn't move
a dot around at negative 0.5, if you go right up here,
that looks like it's around-- this is 2 and 1/2, so this
would be a little bit over 1. It looks like that would be
the slope of the tangent line right over there. Or if we went all the
way to negative 2.25, the slope of the
tangent line looks like it's approximately
30 right over there. So once you have
the curve, you now have the slope of
the tangent line, or you have the
slope of the curve, at every point that
we see over here. So this is just, I thought,
a pretty neat exercise.