help me with this derivative interpretation?

Question

A t a particular location, f(p) is the number of gallons of gas sold when the price is p dollars per gallon.

find and interpret the derivative of the inverse function of 4023.
(f^-1)'(4023)

I got (f^-1)'(4023) = -1/1250

Answer 1

Let v be the volume of gas sold (in gallons).

Then if f(p) = v, the inverse function is (f^-1)(v) = p.
The derivative (f^-1)'(v) is, by definition, dp/dv, that is, the change in price associated with a unit change in volume of gas sold.

The particular equation you give at the end of your question would be interpreted as follows:

When the price of gas is such that 4023 gallons are sold, a price reduction of 1/1250 of a dollar (which is 8/100 of a cent) corresponds to an increase of 1 gallon in the amount of gas sold. In other words, the elasticity of the demand for gas can be measured as 1 gallon per 8/100 cents, or 12.5 gallons per cent, or 1250 gallons per dollar of price reduction.

Answer 2

let f^-1 be g
f gives quantity sold based on the market price (p).

Now, g would give market price on the basis of the quantity sold (q). First derivative of g is the change in price to the change in quantity sold. g'(4023) would indicate how much the price would decline (as the result is negative) when the 4023rd unit is sold.