Q.The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x)=13x^{2}+26x+15$. Find the marginal revenue when $x=7$.

Marginal Revenue: The Revenue from One More Unit

Imagine you run a small stall selling lemonade. You sell 10 glasses at ₹10 each, so total revenue is ₹100. Now you wonder: If I sell one more glass, how much extra money will I actually bring in? That extra money — the change in total revenue from the 11th glass — is marginal revenue.

The Core Intuition

Marginal revenue (MR) answers a practical question: "What happens to total revenue when I increase sales by exactly one unit?"

It is not simply the price of that extra unit. Selling one more might force you to lower the price on all units — especially if you have some market power (like a monopoly or a price-setting firm). In perfect competition, where you're a price-taker, MR equals the market price; but in most real-world scenarios, MR is less than the price.

The Precise Definition

Marginal Revenue is the rate at which total revenue changes as the quantity sold changes. For a firm selling $q$ units with total revenue $R(q)$:

If the demand curve is $p(q)$ (price as a function of quantity), then $R(q) = p(q)\times q$, and by the product rule:

This says marginal revenue equals the price of the current unit plus the effect that selling one more has on the price of all previous units (since $dp/dq$ is usually negative — to sell more, you lower the price).

A Concrete Example

Suppose a monopolist faces demand $p = 100 - 2q$ (price drops ₹2 per extra unit sold), so total revenue is $R(q) = q(100-2q) = 100q - 2q^2$.

The instantaneous marginal revenue is the derivative:

At $q = 10$: $MR(10) = 100 - 40 = 60$. Checking with the product-rule formula: $p(10) = 100 - 20 = 80$, and $dp/dq = -2$, so $MR = 80 + 10\times(-2) = 60$ — the two methods agree.

Compare this to the discrete one-unit change: at $q=10$, $R = 80\times10 = 800$; at $q=11$, $p = 78$ and $R = 78\times11 = 858$, so the actual revenue gained from the 11th unit is $858 - 800 = 58$.