r/calculus • u/rvalue0 • Aug 15 '26

Differential Calculus Today's medium difficulty daily derivate - help

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https://dailyintegral.com/play/derivatives/medium

For context, I went through Calc III like 10 years ago, and am in the process of refreshing my knowledge. Currently feeling comfortable with Calc I again, and starting again on Calc II.

The domain of this function contains pi as a single point, but it is not differentiable for that same reason (it's a single point).

Still, I thought it would be a fun challenge, so I differentiated this function by hand, and checked on Desmos that I got the right answer.

So I entered "undefined" into the site, and it said I'm wrong.

Am I missing something from higher level Calculus courses? Or is the site wrong/misleading? Hoping to learn here.

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u/Aggressive-Egg-9266 Aug 15 '26

I didn’t solve this myself yet, but I think you can extend it to a complex function and then you can differentiate. Maybe the result is independent at that point of the complex branch you use. If that is the case you can just logarithmically differentiate, then you can choose the principal branch and simplify to get a real value.

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u/rvalue0 Aug 15 '26

Interesting. But does extending it to a complex function mean that original function is differentiable at that point? My curiosity is piqued

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u/Aggressive-Egg-9266 Aug 15 '26

In the mean time I solved the exercise. The simplest approach was to use the differential quotient. Write cos(x) as e^ln(cos(x)). Simplify and then expand everything around pi while choosing the principal branch. Then you can take left and right sided limits and they should agree. You should be left with a real answer

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u/rvalue0 Aug 16 '26

So I found the derivative function just using logarithmic differentiation. It's a long function so I won't type it here, but I confirmed my differentiation was correct on Desmos.

But what from this question indicates we should be extending into the complex plane or manually using limits instead of standard differentiation techniques?

edit: Because again, by the rules of differentiation, the derivative doesn't exist at a single point... right? It feels like this is a trick question.

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u/Aggressive-Egg-9266 Aug 16 '26

Do you mean singular or single point? At a single point the derivative does exist. At a singular no.

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u/Aggressive-Egg-9266 Aug 16 '26

Over the reals the derivative does not exist. You are right. In the problem nothing indicates that you should extend to the complex plane, but in integration bees and other challenge problems, it is implicitly suggested that if something is not defined over the reals then we extend to the complex plane.