Question

Find the derivatives of the functions. g(x)=\ln ∣3x−1∣

Derivatives
ANSWERED
asked 2021-05-23

Find the derivatives of the functions. \(g(x)=\ln ∣3x−1∣\)

Answers (1)

2021-05-24
Recall the generalized rule for the derivative of a natural logrithm of an absolute value:
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\ln}{\left|{u}\right|}\right]}={\frac{{{1}}}{{{u}}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}\)
In this exercise, we want to find the derivative of
\(\displaystyle{g{{\left({x}\right)}}}={\ln}{\left|{3}{x}-{1}\right|}\)
To make use of the rule in Step 1, denote the expression inside the absolute value by u, i.e.
u=3x-1
Using the above-mentioned rule, we get
\(\displaystyle{g}'{\left({x}\right)}={\frac{{{1}}}{{{u}}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}={\frac{{{1}}}{{{3}{x}-{1}}}}{\frac{{{1}}}{{{\left.{d}{x}\right.}}}}{\left({3}{x}-{1}\right)}={\frac{{{3}}}{{{3}{x}-{1}}}}\)
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