Let us begin this section with a brief discussion of the mathematical definition of a derivative.
I think you probably know what a derivative is, but in case you don't remember exactly, this is how
we define a derivative mathematically.
So we start from a function f of X, for example, this blue function here with it, which is a third
order polynomial.
And then we are asking about the derivative at a certain position.
For example, here at X equal minus three point five, which is this red dot here that corresponds to
the blue curve.
And then what we can do is we can go along the positive x direction and to find another point.
For example, here at X equal to minus one where we have a difference in X direction of two point five.
So then we have a difference in the Y coordinates of these two points and a difference in the X coordinate
and the two points.
And this defines the slope of this orange curve here.
And this is in a very bad approximation, the derivative of the function at the Red Point.
And this derivative, this approximation of the derivative gets much better when we decrease the distance
in the X direction.
So this means when we try to move the Orange Point towards the Red Point, and this is what I have shown
here.
So imagine that like in the video, you take this Orange Point and you drag it along the curve and then
you will find that this orange straight line here is linear function, but in the end, look like this.
And so the slope of this function is really the derivative of F of X at the position of X equal to minus
three point five.
And then mathematical terms.
We can write this down as the limit of some number h, which is the distance in the X direction, and
the limit goes for h to zero.
And then we have this term here, which is basically the difference in the Y coordinates and the difference
in the X coordinates.
So this is exactly the slope of this function.
So there exist also equivalent definitions for this expression in case our function is continuous,
which it will basically always be in physics.
So we could also take a backward derivative.
So this means we have two points in the origin point is this time on the left side?
Or we could also do a center or a central point or a central definition for the derivative where we
have two points, one along the positive and one along the negative direction, and we push both of
these points towards the Red Point.
And in all cases, we will end up with this red line, which has a slope and the slope is equal to the
derivative at this point.
So I think so far you already have learned what the derivative is, and I hope you remember how this
is defined.
But in case not, it's really easy.
It's just as limit h to zero.
And then the difference in y divided by the difference in X and in the next lecture, we're going to
talk about the numerical implementations of this first or the derivative.