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Question
Find the marginal demand of a commodity where demand is x and price is y.
y = `(5x + 9)/(2x - 10)`
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Solution
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`
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