Total revenue differentiation/calculus

Question

I'm studying a concept that transforms the total revenue function

$TR = P(Q)\cdot Q$

into

$\frac{dTR}{dQ} = (\text{change in P})\cdot Q + P$

Does anyone has an idea how one got to the other please? thanks!!

Answer 1

You have to apply the $\color{sepia}{\text{product rule}}$: $\left(u(x)\cdot v(x)\right)^{'}=u^{'}(x)\cdot v(x)+u(x)\cdot v^{'}(x)$.

$$\frac{dTR}{dQ}=\frac{dP(Q)}{dQ}\cdot Q+P(Q)\cdot \frac{dQ}{dQ}$$

It is obvious that $\frac{dQ}{dQ}=1$. Thus

$$\frac{dTR}{dQ}=\frac{dP(Q)}{dQ}\cdot Q+P(Q)$$

Answer 2

What we are doing or should be doing is taking the Partial Derivative.

The analysis in Economics is Ceteris Paribus. The effect that a marginal change in quantity has in the total revenue while everything else is constant.

The Price is a function of Quantity.

For simplicity sake Lets say(100 - Q)

So when you multiply the Price(100 - Q) and the Quantity(Q)

You get

$100Q-Q^2$

The derivative(Marginal Revenue) is

100 - 2Q

You probably want to Maximise your total Revenue so set the Marginal Revenue to 0

A Quantity greater than 50 would actually make you lose Revenue. The Second derivative is negative so you see that anything greater than 50 would make the First derivative less than 0 and the First derivative is the Marginal Change in Total Revenue.

To Maximise the Profit the Marginal Cost will Probably not be 0.

Set the Marginal Cost equal to whatever the Marginal Revenue is.

Or create a function of Total Profit whci will be the Difference of Total Revenue and Total Cost.

I did not understand what exactly what you are asking but intuitevely one will not be able to sell any quantity of a product at the same price(unless the elasticity is 0 an inelastic demand to the price) and that the derivative and the elasticity are negative is the observation Law of Price and Demand. As for MR = MC you really will not sell a good unless the price offered to you is greater than the cost of producing it.