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Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 5 + x^{2}e^{–x} + 2x

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#### Solution

y = 5 + x^{2}e^{–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`

= x^{2}.e^{–x}(– 1) + 2xe^{–x} + 2

= – x^{2}e^{–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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