# Partial Derivative Product with variables as functions

## 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?

Last edited:

HallsofIvy
Homework Helper
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?

HallsofIvy