Cem Hayes

2021-05-12

Find the derivatives of the functions. $r\left(x\right)=\mathrm{ln}|x+{e}^{x}|$

Corben Pittman

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

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