Question

When finding the derivative of ln|x|

Why is the derivative 1/x and not 1 divided by the absolute value of x?

Answer 1

\[D[ln(abs(x))]\]
\[D[ln(sqrt(x^2))]*D[sqrt(...)]*D[x^2]\]
\[\frac{1}{sqrt(x^2)}*\frac{1}{2sqrt(x^2)}*2x\]
\[\frac{1}{|x|}*\frac{1}{2|x|}*2x\]
\[\frac{x}{|x||x|}\]
when x is a negative value, we still have a negative slope
\[\frac{-3}{|-3||-3|}=-\frac{1}{3}\]