 Calculus: Derivatives 1 | Taking derivatives | Differential Calculus | Khan Academy

October 7, 2019

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