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Vi Hart's introduction to the derivative by explaining what the derivative is for a car ride---and then what the second and third derivatives correspond to.

3Blue1Brown has an 12-part video series on the "Essence of Calculus." The derivative is discussed from many different perspectives. The rest of the videos relating to derivatives are below.

Here's the second derivative video in the aforementioned series.

Here's the third derivative video in the aforementioned series. It's on means to visualize the chain rule (Theorem 7.13) and the product rule (Theorem 7.11).

Here's the fourth derivative video in the aforementioned series. Among other things, it's on l'Hopital's Rule (Theorem 7.27).

Here's the final derivative video in the aforementioned series.

The open question in the derivative chapter (page 256) asked about why if take the derivative of the π*r*² formula for the area of circle, do you get the formula for the diameter: 2π*r*. And, more importantly, how deep does this correspondence go? Related issues between geometric formulas are discussed in this video.

The devil's staircase is a function which has a derivative of zero nearly everywhere, and yet somehow climbs from (0,0) to (1,1). And here's the crazy thing: It does so *continuously*. This is the topic of Example B.8 in Appendix B, as well as of this video by PBS Infinite Series.

This video, from Numberphile, discusses infinitesmals, which were originally used to define the integral, and prove theorems thereof. This video also touches on some of the interesting history of this development, including the rivalry between Newton and Leibniz.

Check out this great video

Check out this great video

Section 8.8 was about the *measure zero integrability criterion,* and this idea of measure was investigated further in Example B.17 of Appendix B. This idea of measure is important in real analysis and a connection between it and music is discussed in this 3Blue1Brown video.

The final open questions of Chapter 2 (on page 63) deal with the *axiom of choice*. This video talks generally about this axiom, as well as a specific application to the measure of sets, which was discussed in the zero case in Section 8.8. This specific application was also discussed in Fact B.17 of Appendix B.

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