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