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Question
Find the marginal demand of a commodity where demand is x and price is y.
y = `(5"x" + 9)/(2"x" - 10)`
Solution
y = `(5"x" + 9)/(2"x" - 10)`
Differentiating both sides w.r.t.x, we get
`"dy"/"dx" = "d"/"dx"((5"x" + 9)/("2x" - 10))`
`= (("2x" - 10)*"d"/"dx"("5x" + 9) - ("5x" + 9) * "d"/"dx"(2"x" - 10))/(2"x" - 10)^2`
`= ((2"x" - 10)(5 + 0) - (5"x" + 9)(2 - 0))/(2"x" - 10)^2`
`= (5(2"x" - 10) - 2(5"x" + 9))/(2"x" - 10)^2`
`= (10"x" - 50 - 10"x" - 18)/(2"x" - 10)^2`
∴ `"dy"/"dx" = (- 68)/(2"x" - 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)/(2"x" - 10)^2) = (- (2"x" - 10)^2)/68`
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