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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.

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I think you just use the product rule:

$$ \begin{align} dR&=\frac{\partial \{P*Y(P, C)\}}{\partial P}dP+\frac{\partial \{P*Y(P, C)\}}{\partial C}dC\\ &=dP\left(Y(P, C)+P\frac{\partial Y(P,C)}{\partial P}\right)+P\frac{\partial Y(P,C)}{\partial C}dC \end{align} $$

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After further investigating I don't believe the above answer is correct. I agree with the first line but don't follow the application of the product rule seen in the second line. Why would the partial w.r.t C include C. I have something to the effect of dR= dP*(Y(P,C)+P*(DY(P,C)/DP)) + (DY(P,C)/DC)*dC where D is the partial derivative operator. Does this seem correct? – user98167 Oct 3 '13 at 0:51
@user98167 Yes, naturally. Sorry for the confusion. Edited my post now. – hejseb Oct 3 '13 at 5:53
@user98167 In your answer I think you are missing one part though. The last term should be multiplied by P (since you have $P*Y(P, C)$). – hejseb Oct 3 '13 at 5:55

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