In business calculus, if $C\left(x\right)$ is the total cost to produce $x$ units, what is the marginal cost at output level $x$?

• A. The average cost per unit: $\frac{C\left(x\right)}{x}$
• B. The total cost at output level $x$: $C\left(x\right)$
• C. The change in output divided by the change in total cost: $\frac{\Delta x}{\Delta C}$
• D. The derivative of total cost with respect to output: $\frac{dC}{dx}$

Answer: (D)

Verified step by step guidance

1. Understand that the total cost function, \(C(x)\), represents the total cost to produce \(x\) units of a product.
2. Recall that the marginal cost at output level \(x\) is the rate at which the total cost changes as the output changes, which means it measures the cost of producing one additional unit at that level.
3. Recognize that the average cost per unit is given by \(\frac{C(x)}{x}\), but this is different from marginal cost because average cost spreads total cost evenly over all units produced.
4. Note that the change in output divided by the change in total cost, \(\frac{\Delta x}{\Delta C}\), is the reciprocal of what we need; marginal cost involves the change in cost per unit change in output, so we focus on \(\frac{\Delta C}{\Delta x}\).
5. To find the exact marginal cost at output level \(x\), take the derivative of the total cost function with respect to \(x\), which is \(\frac{dC}{dx}\). This derivative gives the instantaneous rate of change of cost with respect to output, representing the marginal cost.