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So welcome back to the cause.

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Let's get started with this new section about derivatives and actually in the previous section, which

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was about interpellation and about serious expansions.

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We have already encountered derivatives before the Taylor expansion, we needed first and also higher

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order derivatives for approximating a function.

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And back then, we have used a very simple version of calculating a derivative.

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And I told you that we will later on explore this in more detail.

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So now's the time.

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I will show you different ways on how to implement a derivative in Python or in numerics generally.

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So there are the methods called forward and backward differences.

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Central differences methods.

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And then we also discussed the so-called Richardson method.

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And the idea of these methods is that when you just have some data points, then you can calculate the

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derivative that describes or the derivative of the function that describes two data points with a higher

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accuracy when you use these methods.

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For example, the central differences method or the Richardson method.

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So maybe this sounds confusing to you.

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How it can you get a better result when you just use the same data?

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But it really works, and I want to show you how it works.

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And even you will explore this yourself because as an exercise in the middle of this section, I will

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give you a dataset where where we will investigate the motion of a car, for example, I give you the

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position of the car at several points of times, and then I ask you to calculate the velocity and the

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acceleration using the different methods that we have discussed before.

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And then you will see that some of these methods are way more accurate than other methods than in the

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end of the course or of the section.

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We will discuss multi-dimensional derivatives.

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So this means we will discuss the concept of gradients, curls and divergences.

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And these are extremely relevant in physics, for example, in the electrodynamics, when we describe

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the Maxwell equations, for example, and we will actually discuss a physical problem in the next section

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where we will combine integrals and these multidimensional derivatives to calculate the magnetic fields

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of a charged wire.