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How to make a mathematical dynamic graph?

I have seen a lot of mathematics teaching videos containing a lot of mathematical dynamic graphs; I think it is very interesting, and it is very attractive!

Such a video can help us have a good understanding of mathematics.

So what tool or software should I use to make such a mathematical dynamic picture and insert it into the video?

As an example of what I mean, I feel that one of the excellent teaching science videos is here, from 3Blue1Brown.

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Somewhere in the comments somebody asked the same question and this link is the answer: https://talkingphysics.wordpress.com/2018/06/11/learning-how-to-animate-videos-using-manim-series-a-journey/

A somewhat easier way but far less powerful would be the program geogebra where you can export animations as gif.

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    $\begingroup$ At an intermediate level, mathematica has some good graph-animation tools. $\endgroup$ – Opal E Jul 29 at 21:26
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The video to which you link is much more than a dynamic graph: It is an entire lecture using many dynamic components, requiring considerable skill with several technologies.

To return to a single dynamic graph, in response to this MO question, Taking “Zooming in on a point of a graph” seriously, I created this little animation for $y=x(x−1)(x+1)$:

           Zoom
I created this animated .gif in Mathematica. Similar graph animations could be made in Geogebra or Cinderella.

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    $\begingroup$ +1 for geogebra, it is very visual which is good if you don't have much of a programming background $\endgroup$ – Shai Jul 31 at 17:51
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Desmos may be less powerful than geogebra, but is worth mentioning in this regard. Graphs on desmos are very easily animated (with sliders), and look great with little modification. You can find much better examples, but here's a very simple one I made for thinking about the tangent line to a curve.

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You mentioned 3Blue1Brown's videos. If you're interested in making similar animations, Grant (the man behind 3B1B) actually released the entire python library he wrote for creating them: manim on github. Obviously, this is very open ended and requires a working knowledge of python to begin with. The other examples of software (Desmos, Mathematica, etc) will generally be much more friendly if you just need to whip something together for a lecture or slide. However, manim+python will open up loads of customization and options.

The subreddit /r/manim has additional examples and resources!

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