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

Cem Hayes

Answered question

2021-05-12

Find the derivatives of the functions.

r (x) = \ln | x + e^{x} |

Corben Pittman

Skilled2021-05-13Added 83 answers

Recall the generalized rule for the derivative of a natural logarithm of an absolute value:

\frac{d}{d x} [\ln | u |] = \frac{1}{u} \frac{d u}{d x}

In this exercise, we want to find the derivative of

r (x) = \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^{'} (x) = \frac{1}{u} \frac{d u}{d x} = \frac{1}{x + e^{x}} \frac{d}{d x} (x + e^{x}) = \frac{1 + e^{x}}{x + e^{x}}

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