Is d/dx of ln(|x|) = 1/x? Proving the Claim

[vikcool812]

I know d/dx of ln(x) is 1/x , x>0.
A website says d/dx of ln(|x|) is also 1/x for x not = 0 .
is that true , i am unable to prove it!

 

[arildno]

Use the chain rule of differentiation.

 

[rzaidan]

Hi vikcool812 :
Note that ln(|x|)=ln(x) when x>0 and
=ln(-x) when x<0
So d/dx (ln(|x|))=d/dx(ln(x)) when x>0 and
d/dx (ln(|x|))=d/dx(ln(-x)) when x<0 therefore
d/dx(ln(x)) =1/x when x>0 by definition of ln(x) and
d/dx(ln(-x))=-1/-x = 1/x when x<0 by chain rule so in both cases we have
d/dx (ln(|x|))=1/x for not x=0
Best Regards
Riad Zaidan

 

[g_edgar]

BEWARE... The formula
$$\frac{d}{dx}\ln(|x|) = \frac{1}{x}$$
is wrong for complex $x$

 

[n!kofeyn]

Remember that when dealing with absolute values, with either derivatives or limits, it is best to directly use the definition of the absolute value. I.e.,
$$|x| = \begin{cases} x &, x\geq 0 \\ -x &, x\leq 0 \end{cases}$$
This is of course what rzaidan used in his solution.

 

[vikcool812]

Thank You ! Everyone for your valuable suggestions .

 

[rzaidan]

Hi g_edgar
THankyou for your hint , and my work was in the real numbers.
Best Regards
Riad Zaidan