Can I differentiate $x$ with respect to $|x|$?

Asked Nov 28 '19 at 14:32

Active Nov 28 '19 at 14:32

Viewed 59 times

First, I naively thought that it would be like this. $$\frac{dx}{d|x|} = \frac{1}{\frac{d|x|}{dx}}$$

Then, however, I had an apprehension that since $|x|$ is not differentiable on $x=0$, this conclusion might be mathematically wrong.

So... can I differentiate $x$ with respect to $|x|$ like this, or do I need some other conditions (such as "$0$ is not included in $x$'s domain" or whatever)?

asked Nov 28 '19 at 14:32

JS S

2

$x$ is not a function of $|x|$, so you would have to define what you mean by $dx/d|x|$ for the inequality to make sense. – Roberto Rastapopoulos Nov 28 '19 at 14:34
2

What function is $x$ of $|x|$? Note that restricting to (connected) domains where $|x|$ is invertible makes the problem trivial since either $|x| = x$ or $|x| = -x$ on such a domain. – Sarvesh Ravichandran Iyer Nov 28 '19 at 14:43

Can I differentiate $x$ with respect to $|x|$?

0 Answers0