# Find derivative of gas pressure formula

I did really well with derivatives until I hit this question. I'm not sure how to treat the different variables which are considered constants. I keep getting zero. Can anyone give me a pointer or just start it for me so I can see how to begin and take it from there?

If gas in a cylinder is maintained at a constant temperature​ T, the pressure P is related to the volume V by a formula of the form $$P=\frac{nRT}{V-nb}-\frac{an^2}{V^2}$$, in which​ a, b,​ n, and R are constants. Find $$\frac{\partial P}{\partial V}$$.

• What have you tried? Have you tried using quotient rule? – Alerra Sep 21 '18 at 20:21

Maybe expressing it like this will help:

$$P=nRT(V-nb)^{-1}-an^2V^{-2}$$

Then you can apply the power rule and chain rule, holding everything on the RHS constant except $$V$$:

$$\frac{\partial P}{\partial V} = -nRT(V-nb)^{-2}\left(\frac{\partial}{\partial V}(V-nb)\right)+2an^2V^{-3}\left(\frac{\partial V}{\partial V}\right),$$

where the derivatives in parentheses evaluate to $$1$$.

So:

$$\frac{\partial P}{\partial V} = -\frac{nRT}{(V-nb)^2}+\frac{2an^2}{V^3}$$

• Wonderful...can't believe how easy that is once you know...thanks! – blizz Sep 21 '18 at 20:28
• @blizz I filled in some steps. – Geremia Sep 21 '18 at 20:30