Q.The total cost $C(x)$ associated with the production of $x$ units of an item is given by $C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000$. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Marginal Cost: The Cost of One More

Imagine you run a small bakery. Every morning you bake 50 loaves; your total cost — rent, flour, yeast, electricity, your own time — comes to ₹2,000, an average of ₹40 per loaf.

Now a customer asks for one more loaf. What does baking that 51st loaf actually cost? Not ₹40 — the rent doesn't change and your time is already paid; the oven is already hot. You need only a little more flour, yeast, and electricity: maybe ₹15.

That ₹15 is the marginal cost.

The Core Intuition

Marginal cost answers one question: "If I produce one more unit, how much does my total cost increase?"

It is not the average cost. It is not the total cost. It is the change in total cost when output changes by exactly one unit.

The Precise Definition

Let $TC(Q)$ be the total cost of producing $Q$ units. The marginal cost of the $Q$-th unit — of increasing output from $Q-1$ to $Q$ — is:

When output can change continuously, we use the derivative:

Why It Matters

Marginal cost is the decision-maker's cost. When a firm asks "Should I produce one more unit?", the answer depends on whether the marginal revenue exceeds the marginal cost. If yes, produce it; if no, stop.

This is why marginal cost typically:

• Falls initially — fixed costs are spread out and workers specialise.
• Rises eventually — because of diminishing returns (more workers in a fixed kitchen get in each other's way).

A Quick Example