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प्रश्न
Differentiate the following w.r.t.x. :
y = `(x"e"^x)/(x + "e"^x)`
उत्तर
Let y = `(x"e"^x)/(x + "e"^x)`
∴ `("d"y)/("d"x) = "d"/("d"x) [(x"e"^x)/(x + "e"^x)]`
= `((x + "e"^x) "d"/("d"x) (x"e"^x) - x"e"^x "d"/("d"x) (x + "e"^x))/(x + "e"^x)^2`
= `((x + "e"^x) [x "d"/("d"x) ("e"^x) + "e"^x "d"/("d"x) (x)] - x"e"^x (1 + "e"^x))/(x + "e"^x)^2`
= `((x + "e"^x) [x("e"^x) + "e"^x (1)] - x"e"^x - x"e"^(2x))/(x + "e"^x)^2`
= `(x^2"e"^x + x"e"^x + x"e"^(2x) + "e"^(2x) - x"e"^x - x"e"^(2x))/(x + "e"^x)^2`
= `(x^2"e"^x + "e"^(2x))/(x + "e"^x)^2`
= `("e"^x (x^2 + "e"^x))/(x + "e"^x)^2`
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