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Question
Find the derivative of the following w. r. t. x. : `(xe^x)/(x+e^x)`
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Solution
Let y = `(x"e"^x)/(x + "e"^x)`
Differentiating w.r.t. x, we get
`dy/dx = d/dx((x"e"^x)/(x + "e"^x))`
= `((x + "e"^x)d/dx(x"e"^x) -(x"e"^x)d/dx(x + "e"^x))/(x + "e"^x)^2`
=`((x + "e"^x)[xd/dx("e"^x) + "e"^xd/dx(x)] - x"e"^x(d/dx(x) + d/dx("e"^x)))/(x + "e"^x)^2`
= `((x + "e"^x)[x"e"^x + "e"^x(1)] - x"e"^x(1 + "e"^x))/(x + "e"^x)^2`
=`((x + "e"^x)(x"e"^x + "e"^x) - x"e"^x(1 + "e"^x))/(x + "e"^x)^2`
= `((x + "e"^x)"e"^x(x + 1) - x"e"^x(1 + "e"^x))/(x + "e"^x)^2`
= `("e"^x[(x + "e"^x)(x + 1) - x(1 + "e"^x)])/(x + "e"^x)^2`
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