Chapter 3, Problem 39RE

Revenue Function The revenue, $R$, that a company receives is a function of the weekly sales, $x$. Also, the sales level, $x$, is a function of the weekly advertising expenditures, $A$, and $A$, in turn, is a varying function of time.

Write the derivative symbols for the following quantities: (i) rate of change of revenue, $R$, with respect to advertising expenditures; (ii) time rate of of change of advertising expenditures; (iii) marginal revenue; (iv) rate of change of sales with respect to advertising expenditures. Select your answers from the following:

$\frac{dR}{dx},\frac{dR}{dt},\frac{dA}{dt},\frac{dA}{dR},\frac{dA}{dx},\frac{dx}{dA},$ and $\frac{dR}{dA}$.

Write a type of chain rule that expresses the time rate of change of revenue, $\frac{dR}{dt}$, in terms three derivatives described in part (a).