[MUSIC] So in order to define
basic speed of change, we need to think about
the most simple example here. And the simplest example is actually
what we've seen for the average speed. It's a case of a straight line. If we going all the way with
the same average speed, then we know actually what instantaneous,
which is constant, speed of change. So let us extrapolate this idea for
the case of arbitrary function. So assume the following puzzle,
that we have some function f and we have some given point a. What is the closest linear approximation, which is the most fittest function for
our case in a given point? Let us drew some keys. For example, like this,
here's our blue function f and our red approximation drawn. That's our given point a,
and well the second point of intersection we will call x for
the sake of simplicity. Is it a good approximation? Well, it depends. From our standpoint of view,
it kind of works. But if we just imagine that we
live only inside this red square, it's kind of actually not
such a good approximation. And we need to move our second
intersection point closer to the point a. So let us assume some
basic definitions here. Firstly, the line that we have
drawn recently is called a secant. It's a line which simply intersects
our graph twice in the close neighborhood of the point a, and
that's not a droid we're looking for. So we've decided that we
need to move our point of intersection x towards our given point a. And the idea here is that,
how close we need to move it? Well, we're kind of expected to move
it infinitely close to the point a, which means that we're taking
the limit of a second plane, which is actually called tangent line. And tangent line here are drawn with
green color is actually our ransom. So what about the derivative? Derivative stands for
the speed of change, and the speed of change in our case is slope. So we need to simply define
the slope of our green line. In order to do so,
we define slope of every secant line, which is as a fraction between the change
of function and the change of argument. And then we take the case
where x approaches a. Here our basic definition of
the derivative, congrats. [MUSIC]