Q.The total cost $C(x)$ in Rupees, associated with the production of $x$ units of an item is given by $C(x) = 0.005\,x^3 - 0.02\,x^2 + 30x + 5000$. 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.

Concept: Marginal Cost — the derivative of the total cost function with respect to output.

Step 1: Differentiate $C(x)$ to get the marginal cost function $MC(x)$.

Step 2: Substitute $x = 3$ into $MC(x)$.

$MC(3) = 0.015(9) - 0.04(3) + 30$ …

Why marginal cost?

In economics, "marginal" always means the rate of change — the extra cost of producing one more unit when you're already at a certain output level. For a continuous cost function, that's exactly the derivative $\frac{dC}{dx}$. The problem explicitly says "instantaneous rate of change", so we differentiate, not use average cost.

Step-by-step

1. Write down the cost function

2. Differentiate term by term

   • Derivative of $0.005x^3$: $0.005 \cdot 3x^2 = 0.015x^2$
   • Derivative of $-0.02x^2$: $-0.02 \cdot 2x = -0.04x$
   • Derivative of $30x$: $30$
   • Derivative of constant $5000$: $0$

   So the marginal cost function is

3. Evaluate at $x = 3$

$= 0.015 + 30$ …

Method: Finding a Marginal Quantity from a Given Total Function

This method applies to any "marginal ___" problem in economics-flavoured calculus questions — marginal cost, marginal revenue, marginal profit — where "marginal" always means the derivative of the corresponding total function with respect to the quantity produced or sold.

Steps

Step 1: Recognise that "marginal" means "derivative"

Whenever a question defines marginal cost (or revenue, or profit) as "the instantaneous rate of change of total cost (or revenue, or profit) at any level of output," it is directly telling you to differentiate — no other technique is needed.

Step 2: Differentiate the total function term by term

For $C(x)=0.005x^3-0.02x^2+30x+5000$, apply the power rule to each term:

Note that the constant term (the fixed cost, ₹5000) differentiates to zero — fixed costs never affect the marginal cost, since they don't change as output changes.

Step 3: Substitute the specific output level asked for

Evaluate $MC(x)$ at the given $x$ (here $x=3$) by direct substitution — this is a plain function evaluation, not a further differentiation.

Step 4: State the answer with the correct economic units …

Common Mistakes

Mistake 1: Confusing marginal cost with average cost

A student computes $\frac{C(3)}{3}$ (average cost per unit at $x=3$) instead of $C'(3)$. Why it's wrong: marginal cost is specifically defined in the question as the instantaneous rate of change of total cost — a derivative — not the cost spread evenly over all units produced so far, which is a completely different (and much larger, since it includes the fixed cost) number. Correct approach: differentiate $C(x)$ to get $MC(x)=C'(x)$, then evaluate at $x=3$; never divide $C(x)$ by $x$ for a "marginal" question.

Mistake 2: Forgetting that the constant term contributes zero to the derivative

A student might carry the $5000$ (or some leftover trace of it) into the differentiated marginal cost function. Why it's wrong: a constant total (like fixed rent or setup cost) doesn't change as output changes, so its rate of change is exactly zero — it correctly disappears from $MC(x)$, and including it anywhere in the marginal cost is a sign the differentiation step was done incorrectly. Correct approach: differentiate term by term and confirm the constant term always vanishes. …