Find the derivatives of the functions. r(x)=ln|x+ex|

Recall the generalized rule for the derivative of a natural logarithm of an absolute value:
${\displaystyle \frac{d}{dx}\left[\ln \left|u\right|\right]=\frac{1}{u}\frac{du}{dx}}$
In this exercise, we want to find the derivative of
${\displaystyle r\left(x\right)=\ln |x+e^x|}$
To make use of the rule in Step 1, denote the expression inside the absolute value by u, i.e.
${\displaystyle u=x+e^x}$
Using the above-mentioned rule, we get
${\displaystyle r^{\prime }\left(x\right)=\frac{1}{u}\frac{du}{dx}=\frac{1}{x+e^x}\frac{d}{dx}(x+e^x)=\frac{1+e^x}{x+e^x}}$