Q.The cost (in rupees) of producing x items in factory, each day is given by 𝐶(𝑥) = 0.00013𝑥3 + 0.002 𝑥2 + 5𝑥 + 2200 Find the marginal cost when 150 items are produced.

Concept: Marginal Cost — the rate of change of total cost with respect to the number of items produced, i.e. $C'(x)$.

Step 1: Differentiate $C(x)$ term by term.

$C(x) = 0.00013x^3 + 0.002x^2 + 5x + 2200$

$C'(x) = 0.00039x^2 + 0.004x + 5$

Step 2: Substitute $x = 150$ into $C'(x)$.

$C'(150) = 0.00039(150)^2 + 0.004(150) + 5$

Step 3: Compute stepwise.

$150^2 = 22500$ …

Marginal cost is the derivative of the total cost function. Given

differentiate term by term:

$C'(x) = 3(0.00013)x^2 + 2(0.002)x + 5 = 0.00039x^2 + 0.004x + 5.$ …

Method: Computing a Marginal Quantity at a Given Level

Any question that asks for the "marginal cost," "marginal revenue," or similar at a specific production level $x_0$ is asking for the derivative of the corresponding total function, evaluated at that point.

Steps

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

For a total quantity described by a function $T(x)$ (total cost, total revenue, etc.), the marginal quantity is:

This is the instantaneous rate of change of the total with respect to the number of units.

Step 2: Differentiate the total function term by term

Apply the power rule to each term of the given polynomial (or other) function:

Keep every coefficient exact — do not round until the final numerical step. …

Common Mistakes

Mistake 1: Confusing marginal cost with average cost

Marginal cost is $C'(x)$, the instantaneous rate of change of total cost — not $\dfrac{C(x)}{x}$, the average cost per item. Dividing $C(150)$ by $150$ gives a completely different (and wrong) number for this question.

Mistake 2: Power-rule slip on the cubic term …