In business calculus, if $R\left(x\right)$ is the total revenue from selling $x$ units, what is the best definition of marginal revenue?

• A. The derivative of total revenue with respect to quantity, $\frac{dR}{dx}$, representing the additional revenue from selling one more unit (approximately for small changes).
• B. The total revenue divided by the number of units sold, $\frac{R\left(x\right)}{x}$.
• C. The derivative of price (demand) with respect to quantity, $\frac{dp}{dx}$.
• D. The total revenue minus total cost, $R\left(x\right)-C\left(x\right)$.

Answer: (A)

Verified step by step guidance

1. Understand that total revenue, denoted as \(R(x)\), represents the total income from selling \(x\) units of a product.
2. Recall that marginal revenue is concerned with how total revenue changes as the quantity sold changes, specifically the additional revenue from selling one more unit.
3. Express this concept mathematically by finding the rate of change of total revenue with respect to quantity, which is the derivative \(\frac{\mathrm{d}R}{\mathrm{d}x}\).
4. Interpret \(\frac{\mathrm{d}R}{\mathrm{d}x}\) as the approximate additional revenue generated by increasing sales by a very small amount, ideally one unit.
5. Distinguish marginal revenue from other concepts such as average revenue \(\frac{R(x)}{x}\), the derivative of price with respect to quantity \(\frac{\mathrm{d}p}{\mathrm{d}x}\), and profit \(R(x) - C(x)\), to clarify its unique definition.