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Find rate of change of demand (x) of a commodity with respect to its price (y) if y = `(3x + 7)/(2x^2 + 5)`

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#### 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 + 10))/(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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