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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.
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