Q.The total cost (in rupees) of producing $x$ units of a good is $C(x) = x^{3} - 15x^{2} + 100x + 200$. Find the marginal cost function $MC(x)$ and evaluate the marginal cost when $x=5$ units.

Marginal cost is the derivative of the total cost function, found term by term using the power rule.\n> [!TLDR]\n> $MC(x)=C'(x)$; differentiate term by term, then substitute $x=5$.\n\n$MC(x)=3x^2-30x+100$. $MC(5)=75- …

By definition (§6, §9), the marginal cost function is the derivative of the total cost function: $MC(x)=\dfrac{dC}{dx}$.

Differentiate $C(x)=x^3-15x^2+100x+200$ term by term using the power rule (§7):

Evaluate at $x=5$:

So the marginal cost at an output of $5$ units is ₹25 — the approximate extra cost of producing the $6^{\text{th}}$ unit. …

Discrete comparison check (shown above): computing $C(6)-C(5)=26$ directly from the cost function lands very close to the derivative-based marginal cost of $25$ at $x=5$, consistent with marginal cost being the ins …

A common mistake is evaluating $C(5)$ itself and calling that the marginal cost — marginal cost is the DERIVATIVE $C'(x)$ evaluated at $x=5$, a rate, not the …