Find the marginal demand of a commodity where demand is x and price is y. y = x + 2x2+1 - Mathematics and Statistics

Sum

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

y = ("x + 2")/("x"^2 + 1)

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)

Concept: Derivatives of Inverse Functions
Is there an error in this question or solution?
Chapter 3: Differentiation - Exercise 3.2 [Page 92]

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