1. ## [SOLVED] Quotient Differentiation

Hey I have a question. I've had to differentiate something but the answer in the back of the book says otherwise.
Which one is right? And If I'm wrong can you tell me how to do it?

The first equation is the original f(x)
The second is the answer from the book f '(x)

Thanks!

2. Hello,

Originally Posted by Evales
Hey I have a question. I've had to differentiate something but the answer in the back of the book says otherwise.
Which one is right? And If I'm wrong can you tell me how to do it?

The first equation is the original f(x)
The second is the answer from the book f '(x)

Thanks!
Yours should be $\frac{1}{e^x+x}-\frac{(e^x+1)x}{(e^x+x)^2}$

It would have been better if you had provided your working, but it doesn't matter

$f(x)=\frac{x}{e^x+x}=\frac uv$

\begin{aligned} f'(x)&=\frac{u'v-uv'}{v^2} \\
&=\frac{1 \cdot (e^x+x)-x(e^x+1)}{(e^x+x)^2} \ {\color{red}(1)} \\
&=\frac{e^x+x}{(e^x+x)^2}-\frac{x(e^x+1)}{(e^x+x)^2} \\
&=\frac{1}{e^x+x}-\frac{x(e^x+1)}{(e^x+x)^2} \end{aligned}

From (1), we get :

$=\frac{e^x+x-xe^x-x}{(e^x+x)^2}=\frac{e^x-xe^x}{(e^x+x)^2}$
which is what your book gave.

Is it better ?

3. Originally Posted by Evales
Hey I have a question. I've had to differentiate something but the answer in the back of the book says otherwise.
Which one is right? And If I'm wrong can you tell me how to do it?

The first equation is the original f(x)
The second is the answer from the book f '(x)
$f(x) = \frac{x}{e^x + x}$
$f'(x) = \frac{(1)(e^x + x) - (x)(e^x + 1)}{(e^x + x)^2}$