Q.The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x + 5$. Find the marginal revenue, when $x = 5$, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant.

Concept: Marginal Revenue Calculation — Marginal revenue is the instantaneous rate of change of total revenue, i.e., the derivative $R'(x)$.

Step 1: Differentiate $R(x) = 3x^2 + 36x + 5$ with respect to $x$:

Step 2: Evaluate at $x = 5$: …

Concept First: What is Marginal Revenue?

Marginal revenue answers a practical question: If I sell one more unit right now, how much extra money will I bring in? In calculus, "at an instant" means we want the derivative — the slope of the revenue function at a specific point. Unlike average revenue (which looks at a range), marginal revenue zooms in on the exact moment you sell that 6th unit when you're already at 5 units.

The function $R(x) = 3x^2 + 36x + 5$ gives total revenue in rupees for $x$ units sold. The constant 5 might represent a fixed setup fee or minimum revenue — but when we differentiate, constants vanish because they don't change as you sell more units.

Step-by-Step Solution

1. Identify what we need.

We want the rate of change of $R$ with respect to $x$ at $x = 5$. That's $R'(5)$.

2. Differentiate $R(x)$ term by term.

$R(x) = 3x^2 + 36x + 5$

• Derivative of $3x^2$: bring down the 2, multiply by 3 → $6x$
• Derivative of $36x$: $36$ (since derivative of $x$ is 1)
• Derivative of $5$: $0$ (constant rule)

So $R'(x) = 6x + 36$.

3. Plug in $x = 5$. …

Method: Computing an Instantaneous Rate (Marginal Value) by Differentiation

This method applies whenever a question gives a total quantity (revenue, cost, profit, volume, etc.) as a function of one variable and asks for its rate of change — often called the "marginal" value — at a specific instant.

Steps

Step 1: Recognise that "rate of change at an instant" means "derivative."

Any phrase like "marginal revenue," "marginal cost," or "rate of change of ... with respect to ..." at a particular value of the variable is asking for the derivative of the given function evaluated at that point — not the function's own value, and not an average change over a range.

Step 2: Differentiate the given function using the standard rules.

Apply the power rule term by term to whatever polynomial (or other differentiable expression) is given:

$\frac{d}{dx}(ax^n) = n a x^{n-1}, \qquad \frac{d}{dx}(\text{constant}) = 0$ …

Common Mistakes

Mistake 1: Evaluating the revenue function itself instead of its derivative.

Why it's wrong: computing $R(5) = 3(25)+36(5)+5 = 260$ answers "what is the total revenue at $x=5$," not "what is the marginal (rate of change of) revenue" — these are different quantities. Correct approach: always differentiate first to get $R'(x)$, then substitute the given value into $R'(x)$, never into $R(x)$.

Mistake 2: Computing an average rate of change instead of the instantaneous rate. …