# Taking a derivative?

1. Sep 7, 2009

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

I have a system that is defined by the equation:

u = Av-2exp(s/R)

I'm looking for the final temperature of the system knowing that as the system changes, pressure will be halved and (s/R) will be a constant.

The equation which relates energy, temperature, and pressure is du = Tds - Pdv

How do I take the derivative of Av-2exp(s/R) to get it in the form Tds - Pdv?

2. Relevant equations

$$\partial$$u$$\partial$$s=T
$$\partial$$u$$\partial$$v=-P
du = $$\partial$$u$$\partial$$s ds + $$\partial$$u$$\partial$$v dv

3. The attempt at a solution

Do I take the partial derivative of the whole thing for v, then add another term that is the partial derivative of the whole thing for s?

-2Av-3exp(s/R) + partial derivative for exp(s/R) with respect to s = du?

And, when you're taking the derivative of exp(s/R), does it come out something like: 1/Rexp(s/R)?

2. Sep 7, 2009

### Staff: Mentor

I think what you're looking for is what's called the total differential, du.

$$du = \frac{\partial u}{\partial s}~ds~+~\frac{\partial u}{\partial v}~dv$$