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प्रश्न
Find the marginal demand of a commodity where demand is x and price is y.
y = `"x"*"e"^-"x" + 7`
योग
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उत्तर
y = `"x"*"e"^-"x" + 7`
Differentiating both sides w.r.t.x, we get
`"dy"/"dx" = "d"/"dx"("x"*"e"^-"x" + 7)`
`= "d"/"dx"("x"*"e"^-"x") + "d"/"dx" (7)`
`= "x" * "d"/"dx"("e"^-"x") + "e"^-"x" * "d"/"dx"("x") + 0`
`= "x" * "e"^-"x" * "d"/"dx"(- "x") + "e"^-"x"(1)`
`= "x" * "e"^-"x" (- 1) + "e"^-"x"`
`= "e"^-"x"(- "x" + 1)`
∴ `"dy"/"dx" = (- "x" + 1)/"e"^"x"`
Now, by derivative of inverse function, the marginal demand of a commodity is
`"dx"/"dy" = 1/("dy"/"dx")`, where `("dy"/"dx") ne 0`
i.e. `"dx"/"dy" = 1/((- "x" + 1)/"e"^"x") = "e"^"x"/(1 - "x")`
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