Lets say I have a function:$$\nu=\frac{RT}{P}+B_{p}(T)RT$$ and I am trying to find $\left(\frac{\partial \nu}{\partial T}\right)_{P}$. I understanding that the partial derivative of the first term is just $\frac{R}{P}$ but the second term has two terms that depend on T. Do I use the product rule like regular derivatives? If so, I think the answer would look something like this:
$$\left(\frac{\partial \nu}{\partial T}\right)_{P}=\frac{R}{P}+RT \frac{\partial B_{P}(T)}{\partial T}+RB_{P}(T)$$
Am I correct in saying this or does the product rule not apply to partial derivatives like I was thinking.