# Partial Derivative Product with variables as functions

1. Nov 21, 2012

### Estane

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

(∂/∂p) (v(p)p(x,t)) = v(p) + (∂v/∂p)p

3. 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: Nov 21, 2012
2. Nov 21, 2012

### HallsofIvy

Staff Emeritus
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.

3. Nov 21, 2012

### Estane

So if I'm understanding this correctly, all the derivatives should be ordinary rather than partial in this case?

4. Nov 21, 2012

### HallsofIvy

Staff Emeritus
Yes.

5. Nov 21, 2012

### Estane

Ok, thank you for your help.