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Question
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`
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Solution
y = `(5x + 7)/(2x - 13)`
Differentiating both sides w.r.t. x, we get
`("d"y)/("d"x) = "d"/("d"x) ((5x + 7)/(2x - 13))`
= `((2x - 13)*"d"/("d"x) (5x + 7) - (5x + 7)*"d"/("d"x)(2x - 13))/(2x - 13)^2`
= `((2x - 13)(5 xx 1 + 0) - (5x + 7)(2 xx 1 - 0))/(2x - 13)^2`
= `((2x - 13)(5) - (5x + 7)(2))/(2x - 13)^2`
= `(10x - 65 - 10x - 14)/(2x - 13)^2`
∴ `("d"y)/("d"x) = (-79)/(2x - 13)^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 `"dy"/"dx" ne 0`
i.e. `("d"x)/("d"y) = 1/((- 79)/(2x - 13)^2)`
`= (-(2x - 13)^2)/79`
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