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Question
Find rate of change of demand (x) of a commodity with respect to its price (y) if y = `(3x + 7)/(2x^2 + 5)`
Solution
y = `(3x + 7)/(2x^2 + 5)`
Differentiating both sides w.r.t. x, we get
`("d"y)/("d"x) = "d"/("d"x)((3x + 7)/(2x^2 + 5))`
= `((2x^2 + 5)*"d"/("d"x)(3x + 7) - (3x + 7)*"d"/("d"x)(2x^2 + 5))/(2x^2 + 5)^2`
= `((2x^2 + 5)(3 + 0) - (3x + 7)(4x + 0))/(2x^2 + 5)^2`
= `(6x^2 + 15 - 12x^2 - 28x)/(2x^2 + 5)^2`
= `(-6x^2 - 28x + 15)/(2x^2 + 5)^2`
Now, by derivative of inverse function, the rate of change of demand (x) w.r.t. price (y) is
`("d"x)/("d"y) = 1/(("d"y)/("d"x))`, where `("d"y)/("d"x) ≠ 0`
i.e, `("d"x)/("d"y) = 1/((-6x^2 - 28x + 15)/(2x^2 + 5)^2`
= `(2x^2 + 5)^2/(- 6x^2 - 28x + 15)`
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