Partial Derivative Product with variables as functions

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Homework Help Overview

The discussion revolves around the application of the product rule in the context of taking partial derivatives of a function product, specifically when the variable of differentiation is a function itself.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to apply the product rule to a function involving a variable that is itself a function of other variables, questioning whether this leads to complications. Some participants discuss the distinction between ordinary and partial derivatives in this context.

Discussion Status

Participants are exploring the implications of using ordinary versus partial derivatives in the context of the product rule. There is a clarification that the derivatives should be treated as ordinary in this specific case, but no consensus on the broader implications has been reached.

Contextual Notes

The original poster is grappling with the nuances of differentiation when variables are functions of other variables, which may affect their understanding of derivative types in this scenario.

Estane
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Homework Statement



I'm trying to understand how a certain substitution can be made with regards to taking the partial derivative of a function product when the variable I am differentiating by is a function itself.

Homework Equations



(∂/∂p) (v(p)p(x,t)) = v(p) + (∂v/∂p)p

The Attempt at a Solution



My first thought was to use the product rule, but that seems to fall down with the function p(x,t) there. Am I missing something obvious here?
 
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Because the derivative is with respect to p, the fact that p is a function of x and y is irrelevent. With f(p)= pv(p), by the product rule, df/dp= v+ p dv/dp. Notice that these are ordinary derivatives, not partial derivatives, because f depends on the single variable, p.
 
So if I'm understanding this correctly, all the derivatives should be ordinary rather than partial in this case?
 
Yes.
 
Ok, thank you for your help.
 

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