So welcome back to the cause.
Let's get started with this new section about derivatives and actually in the previous section, which
was about interpellation and about serious expansions.
We have already encountered derivatives before the Taylor expansion, we needed first and also higher
order derivatives for approximating a function.
And back then, we have used a very simple version of calculating a derivative.
And I told you that we will later on explore this in more detail.
So now's the time.
I will show you different ways on how to implement a derivative in Python or in numerics generally.
So there are the methods called forward and backward differences.
Central differences methods.
And then we also discussed the so-called Richardson method.
And the idea of these methods is that when you just have some data points, then you can calculate the
derivative that describes or the derivative of the function that describes two data points with a higher
accuracy when you use these methods.
For example, the central differences method or the Richardson method.
So maybe this sounds confusing to you.
How it can you get a better result when you just use the same data?
But it really works, and I want to show you how it works.
And even you will explore this yourself because as an exercise in the middle of this section, I will
give you a dataset where where we will investigate the motion of a car, for example, I give you the
position of the car at several points of times, and then I ask you to calculate the velocity and the
acceleration using the different methods that we have discussed before.
And then you will see that some of these methods are way more accurate than other methods than in the
end of the course or of the section.
We will discuss multi-dimensional derivatives.
So this means we will discuss the concept of gradients, curls and divergences.
And these are extremely relevant in physics, for example, in the electrodynamics, when we describe
the Maxwell equations, for example, and we will actually discuss a physical problem in the next section
where we will combine integrals and these multidimensional derivatives to calculate the magnetic fields
of a charged wire.