R revenue, P price, Q quantity sold

Revenue R=PQ

increase P by an amount \Delta P ( \Delta is the Greek letter delta, uppercase)

Q will change by \Delta Q and R by \Delta R

R=PQ (1)

R+\Delta R = (P+\Delta P)(Q+\Delta Q)\approx PQ+P\Delta Q+Q\Delta P (2) (expand and neglect the term \Delta P\Delta Q because it is smaller than the other terms)

subtract (1) from (2)

\Delta R = P\Delta Q+Q\Delta P

marginal revenue

MR=\frac{\Delta R}{\Delta Q}=P+Q\frac{\Delta P}{\Delta Q}=P\left(1+\frac{Q}{P}\frac{\Delta P}{\Delta Q}\right) (3)

everything is straightforward up to here.

The `trick’ now is to realize that price elasticity of demand

E_{p}=-\frac{\Delta Q}{Q}/\frac{\Delta P}{P} so that we can write (3) as

MR=P(1-\frac{1}{E_{p}})

If you prefer derivatives to deltas, then

R=PQ

MR=\frac{dR}{dQ}=P+Q\frac{dP}{dQ}=P\left(1+\frac{Q}{P}\frac{dP}{dQ}\right)

and again you can write this in the desired form using price elasticity of demand

thank you so much!