Question

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

Derivatives

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

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
$$\displaystyle{g}'{\left({x}\right)}={\frac{{{1}}}{{{u}}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}={\frac{{{1}}}{{{3}{x}-{1}}}}{\frac{{{1}}}{{{\left.{d}{x}\right.}}}}{\left({3}{x}-{1}\right)}={\frac{{{3}}}{{{3}{x}-{1}}}}$$