Differentiate f(x)=`(8-xe^x)/(x+e^x)`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x)=`(8-xe^x)/(x+e^x)`

The derivative of f(x) can be determined using the quotient rule.

f'(x) = `((8 - x*e^x)'(x + e^x) - (8 - x*e^x)(x + e^x)')/(x + e^x)^2`

=> `((-e^x - xe^x)(x + e^x) - (8 - x*e^x)(1 + e^x))/(x + e^x)^2`

=> `(-x*e^x - x^2*e^x - e^(2x) - xe^(2x) - 8 + x*e^x - 8e^x + xe^(2x))/(x + e^x)^2`

=> `(-x^2*e^x - e^(2x) - 8 - 8e^x)/(x + e^x)^2`

The derivative of f(x) = `(8-xe^x)/(x+e^x)` is f'(x) = `(-x^2*e^x - e^(2x) - 8 - 8e^x)/(x + e^x)^2`

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