Suppose we have a revenue function: $R= P*Y$ where $P=$ price and $Y=$ output and is a function of $P$ and $C$, $Y= Y(P,C)$. How could we write the total differential of $R$ with respect to $P$ and $C$?
Here's where I am at: $$ dR= \frac{\partial R}{\partial P}dP + \frac{\partial R}{\partial C}dC $$
I am stuck trying to determine the partial of $R$ w.r.t. $P$ and $C$. How should I deal with the $P$ that is being multiplied by $Y(P,C)$?
Thanks for the help.
I think you just use the product rule:
$$ \begin{align} dR&=\frac{\partial \{P*Y(P, C)\}}{\partial P}dP+\frac{\partial \{P*Y(P, C)\}}{\partial C}dC\\ &=dP\left(Y(P, C)+P\frac{\partial Y(P,C)}{\partial P}\right)+P\frac{\partial Y(P,C)}{\partial C}dC \end{align} $$