Question

How do you use the quotient rule to find the derivative of `y=(1-x*e^x)/(x+e^x)` ?

Answer 1

`y=f(x)/g(x)=(1-xe^x)/(x+e^x)`

`f'(x)=-(xe^x+e^x)`

`g'(x)=1+e^x`

`y'=(g(x)f'(x)-g'(x)f(x))/((g(x))^2)`

Substitute in the values for `f(x), f'(x), g(x), and g'(x)`

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

FOIL

`=(-x^2e^x-xe^x-xe^(2x)-e^(2x)-[1-xe^x+e^x-xe^(2x)])/(x+e^x)^2`

Distribute the negative

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

Combine like terms

`=(-x^2e^x-e^x-e^(2x)-1)/(x+e^x)^2`