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Question
Find the marginal demand of a commodity where demand is x and price is y.
y = `("x + 2")/("x"^2 + 1)`
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Solution
y = `("x + 2")/("x"^2 + 1)`
Differentiating both sides w.r.t.x, we get
`"dy"/"dx" = "d"/"dx"(("x + 2")/("x"^2 + 1))`
`= (("x"^2 + 1) * "d"/"dx"("x + 2") - ("x + 2") * "d"/"dx" ("x"^2 + 1))/("x"^2 + 1)^2`
`= (("x"^2 + 1)(1 + 0) - ("x + 2")("2x" + 0))/("x"^2 + 1)^2`
`= (("x"^2 + 1)(1) - ("x + 2")("2x"))/("x"^2 + 1)^2`
`= ("x"^2 + 1 - 2"x"^2 - 4"x")/("x"^2 + 1)^2`
∴ `"dy"/"dx" = (1 - "4x" - "x"^2)/("x"^2 + 1)^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/((1 - 4"x" - "x"^2)/("x"^2 + 1)^2) = ("x"^2 + 1)^2/(1 - 4"x" - "x"^2)`
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