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

1 Answer
Dec 17, 2016

((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