Thermal Energy Equation Term - Chain Rule

[kevman90]

Homework Statement

I am going through a derivation of the thermal energy equation for a fluid and am stumped on one of the steps. Specifically, the text I am using converts the term:

P/ρ*(Dρ/Dt)

to:

ρ*D/Dt(P/ρ) - DP/Dt

where:
ρ = density
P = pressure
D/Dt = material derivative

The text says this is done using the chain rule of differentiation but I can't derive it myself. I'm far removed from calculus so maybe I'm missing something simple but any help would be appreciated.

Homework Equations

The Attempt at a Solution

 

[Charles Link]

One of the two expressions you have needs an extra minus sign. Momentarily, I will show the calculus of the second expression with the chain rule... @kevman90 Do you know how to take the derivative of $\frac{d(uv)}{dt}$? It is $u (\frac{dv}{dt}) +v(\frac{du}{dt})$. In this case, $u=P$ and $v=1/\rho$. With the chain rule, $\frac{dv}{dt}=(\frac{dv}{d \rho}) (\frac{d \rho}{dt})$. Do you know how to compute $\frac{d v}{d \rho}$ ? With that, you should be able to process the second expression that you have, but I think you will find that it equals the minus of your first expression.

 

[berkeman]

One of the two expressions you have needs an extra minus sign. Momentarily, I will show the calculus of the second expression with the chain rule...

Sorry @Charles Link -- I was in the process of deleting the OP and warning for not showing enough work. But if you want to give a couple hints, that's probably okay.

 

[kevman90]

This makes sense - didn't think about using the product rule. I will work through it later but I think I've got it. Also my mistake with the minus sign I forgot to include it out in front of the first term. Thanks!