Find rate of change of demand (x) of a commodity with respect to its price (y) if y = 3x+72x2+5 - Mathematics and Statistics

Sum

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 + 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)

Concept: Derivatives of Inverse Functions
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