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I've got a question regarding a price elasticity problem and a partial derivative.

That's what's given for the exercise:

So, first of all we calculate all the demand with the given information. Which is:

And then we come to the actual problem. (4. b) )

How do they get the partial derivative here? I assume that they are using the quotient rule, right? I would be really glad for a step by step explanation.

Thank you so much!