Question

Find the marginal demand of a commodity where demand is x and price is y.

y = `(5x + 9)/(2x - 10)`

Answer 1

y = `(5x + 9)/(2x - 10)`

Differentiating both sides w.r.t. x, we get

`(dy)/(dx) = (d)/(dx)((5x + 9)/(2x - 10))`

`= ((2x - 10) xx (d)/(dx)(5x + 9) - (5x + 9) xx (d)/(dx)(2x - 10))/(2x - 10)^2`

`= ((2x - 10)(5 + 0) - (5x + 9)(2 - 0))/(2x - 10)^2`

`= (5(2x - 10) - 2(5x + 9))/(2x - 10)^2`

`= (10x - 50 - 10x - 18)/(2x - 10)^2`

∴ `(dy)/(dx) = (-68)/(2x - 10)^2`

Now, by derivative of inverse function, the marginal demand of a commodity is

`(dx)/(dy) = (1/(dy))/(dx)`, where `(dy)/(dx) ne 0`.

i.e. `(dx)/(dy) = 1/((- 68)/(2x - 10)^2)` 

= `(- (2x - 10)^2)/68`