Question

How do you differentiate `f(x)=(1-xe^x)/(x+e^x)`?

Answer 1

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

Explanation:

This is a problem that is using the quotient rule
the formula for this that you have to remember is:
`(u^'v-uv^')/v^2`

The first things that you have to do is determining the derivatives of each of the functions that you have.

`u=1-xe^x->u^'=e^x+xe^x` (remembering that you have to do the product rule in the process `u^'v+uv^'` and that the derivative of a constant = 0)
`v=x+e^x->v^'=1+e^x`

Then you can now just plug all of your u and v and their derivatives in their proper spot and you got your answer:

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

Usually on exams or tests, they will not ask you not to simplify but if you do need to simplify, factor the `e^x` on the top side. And remember when simplifying, never do anything to the bottom