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Question
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 5 + x2e–x + 2x
Solution
y = 5 + x2e–x + 2x
Differentiating both sides w.r.t. x, we get
`("d"y)/("d"x) = "d"/("d"x) (5 + x^2"e"^(-x) + 2x)`
= `"d"/("d"x)(5) + "d"/("d"x)(x^2"e"^-x) + "d"/("d"x)(2x)`
= `0 + x^2*"d"/("d"x)("e"^-x) + "e"^(-x)*"d"/("d"x)(x^2) + 2`
= `x^2*"e"^-x*"d"/("d"x)(-x) + "e"^(-x)*2x + 2`
= x2.e–x(– 1) + 2xe–x + 2
= – x2e–x + 2xe–x + 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"y)/("d"y) = 1/((-x^2"e"^(-x) + 2x"e"^(-x) + 2))`
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