Differentiating f(x)^g(x)

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My favourite mathematical "YouTutor" is Steve Chow, the guy behind the superlative blackpenredpen collection of mathematics videos on YouTube. Steve has developed a way of explaining proofs using two or more pen colours, hence the name of his YouTube channel. He holds multiple marker pens in one hand while writing on a whiteboard, deftly switching colours so as to highlight the essential differences between each pair of lines in the mathematical exposition. It's a very effective expository technique.

Until recently, my number one blackpenredpen video was this one, where Steve uses logarithmic differentiation and the chain rule to prove the power, product, and quotient rules of derivatives. The reason it's so high on my list is because of the additional bonus content at the end, kind of a "hidden track" type deal, where he goes on to solve for the derivative of a function raised to the power of another function.



Just recently Steve, a keen runner of physical marathons, published a truly marathon mathematical video - my new favourite - in which he performs a hundred and one integration proofs in a single take. I'll let you search for that 6 hour video yourself, and if you feel like sitting through it, may I suggest you limit your accompanying playlist to just these three songs: Steely Dan's "My Old School", Supertramp's "School", and Bowling For Soup's "High School Never Ends". This will help you digest the integration marathon in healthy 15 minute chunks.

Getting back to the bonus differentiation, I remember thinking, this procedure has the effect of subtracting one from the power of one of the functions. Yet it doesn't use the power rule in its own derivation. Aha, I bet this could be used to derive the power rule itself! And since I was looking for a suitable subject for my first foray into maths typesetting, using LaTeX in the Overleaf environment, that became my target.

Here is the resultant PDF. You'll notice the homage to blackpenredpen in some of the text colouring.



The actual Overleaf project which produced this PDF can be viewed here, if you're interested in examining the source (LaTeX is a markup language, just as HTML is a markup language).

This is the first in a series of articles looking at aspects of differentiation. Next time I'll be presenting a neat & simple differentiator, utilising the Expression Trees feature of Microsoft C#.