The Monopolist's Price and Output found a monopoly's output where MR = MC and read its price off the demand curve. With a cost function, an inverse demand $p(q)$ and the The Product Rule, the same rule falls out of one derivative.
Key idea
A single-price monopoly maximizes $\pi(q) = p(q)\,q - C(q)$. The first-order condition is $$MR(q) = p(q) + q\,p'(q) = MC(q).$$ Because $p'(q) < 0$, marginal revenue is below price. The price is then read from the demand curve at that $q$.
Why it works
Revenue is a product, $R(q) = p(q)\cdot q$. The product rule splits its derivative into two pieces. Selling one more unit brings in its price, $p(q)$: the gain. But to sell it the firm must lower the price on all $q$ units it was already selling, by $p'(q)$ each: the loss, $q\,p'(q)$. A competitive firm has $p' = 0$, so for it $MR = p$; a monopoly does not. For linear demand $p = a - bq$, $MR = a - 2bq$: the same intercept and twice the slope.
Worked example: Linear demand, constant cost
$p(q) = 100 - 2q$, $MC = 20$, fixed cost 300.
- $R = 100q - 2q^2$, so $MR = 100 - 4q$.
- $100 - 4q = 20$ gives $q = 20$.
- Price from demand: $p = 100 - 40 = 60$.
- Profit: $(60 - 20)(20) - 300 = 500$.
Answer: $q = 20$, $p = 60$, profit 500.
Worked example: Rising marginal cost
$p(q) = 100 - 2q$, $C(q) = 10q + q^2$.
- $MR = 100 - 4q$ and $MC = 10 + 2q$.
- $100 - 4q = 10 + 2q$ gives $q = 15$.
- $p = 100 - 30 = 70$.
Answer: $q = 15$, $p = 70$.
Worked example: Curved demand
$p(q) = 75 - q^2$, $MC = 27$.
- $R = 75q - q^3$, so $MR = 75 - 3q^2$.
- $75 - 3q^2 = 27$ gives $q^2 = 16$, $q = 4$.
- $p = 75 - 16 = 59$.
Answer: $q = 4$, $p = 59$.
Two checks finish the problem. Second-order: profit must curve down at your $q$, which holds when $MR$ falls faster than $MC$. And shutdown: compare the profit at your $q$ with what the firm gets from producing nothing. One more consequence: a monopoly has no supply curve. Its output depends on the whole shape of demand (through MR), not on the price alone, so two different demand curves can lead it to the same price with different quantities.
Common mistake
Setting price equal to marginal cost (the competitive rule), or differentiating $p(q)\,q$ as just $p'(q)$. In the first example $p = MC$ would give $q = 40$ and a price of 20, which leaves the monopoly's profit at minus the fixed cost. And $\dfrac{d}{dq}[p(q)\,q]$ needs the product rule: $p(q) + q\,p'(q)$. Check: your MR must be below the demand curve at every $q > 0$, and your price must come from the demand curve, not from MR or MC.