TITLE: How to show $\frac{d}{d x}\left(|x|^{1/2}\right)=\frac{x}{2|x|^{3/2}}$?
QUESTION [6 upvotes]: $$\frac{d}{d x}\left(|x|^{\frac{1}{2}}\right)=\frac{x}{2|x|^{\frac{3}{2}}}$$
and also the second derivative $$\frac{d^{2}}{d x^{2}}\left(\sqrt{|x|}\right)=\frac{\delta(x)}{\sqrt{|x|}}-\frac{x^{2}}{4|x|^{\frac{7}{2}}}$$
I know this may seem basic. But I don't get how the Dirac delta (thanks for the comments for correcting me) comes in. Is it just a way someone has decided to write the result for $x >0$, $x<0$ and $x=0$ cleverly in one line. Is it arbitrary as long as the Kronecker term beats the other term?
If anyone has a clean derivation for these I'd be grateful.
EDIT: thanks for a lot of your answers, I've learnt a lot.
Still not entirely sure with the second derivative, the formula is here: https://www.wolframalpha.com/input/?i=second+derivative+of+%7Cx%7C%5E1%2F2, is this wrong? I still think it is probably correct. From taking something like. d/dx($\operatorname{sign}(x) \frac{1}{2|x|^{\frac{1}{2}}} )=1/2 ($$\frac{\delta(x)}{\sqrt{|x|}} -\frac{sign(x)*x}{2|x|^{5 / 2}})$ Don't know why first summand is missin 1/2. I know its distributions, but can't you still differentiate sign(x), you just have to be careful and when evaluating use test functions and integrals

REPLY [2 votes]: According to my calculations the correct answer is
$$
(|x|^{1/2})''
= \lim_{\epsilon \to 0} \left( - \epsilon^{-1/2} \, \delta(x) + \frac14 (1-\chi_{(-\epsilon,\epsilon)}(x)) |x|^{-3/2} \right)
$$
Calculation (sorry for lack of comments):
$$\begin{align}
\langle |x|_+^{1/2}, \varphi(x) \rangle &:= \int_0^\infty x^{1/2} \, \varphi(x) \, dx
= \lim_{\epsilon \to 0} \int_\epsilon^\infty x^{1/2} \, \varphi(x) \, dx \\
\langle |x|_-^{1/2}, \varphi(x) \rangle &:= \int_{-\infty}^0 (-x)^{1/2} \, \varphi(x) \, dx = \int_0^\infty x^{1/2} \, \varphi(-x) \, dx = \langle |x|_+^{1/2}, \varphi(-x) \rangle \\
%\langle |x|^{1/2}, \varphi(x) \rangle &:= \langle |x|_+^{1/2}, \varphi(x) \rangle + \langle |x|_-^{1/2}, \varphi(x) \rangle \\
|x|^{1/2} &:= |x|_+^{1/2} + |x|_-^{1/2}
\end{align}$$
$$\begin{align}
\langle D|x|_+^{1/2}, \varphi(x) \rangle
&= -\langle |x|_+^{1/2}, \varphi'(x) \rangle \\
&= -\lim_{\epsilon \to 0} \int_\epsilon^\infty x^{1/2} \, \varphi'(x) \, dx \\
&= -\lim_{\epsilon \to 0} \left( \left[ x^{1/2} \, \varphi(x) \right]_\epsilon^\infty - \int_\epsilon^\infty \frac12 x^{-1/2} \, \varphi(x) \, dx \right) \\
&= \lim_{\epsilon \to 0} \left( \epsilon^{1/2} \, \varphi(\epsilon) + \int_\epsilon^\infty \frac12 x^{-1/2} \, \varphi(x) \, dx \right) \\
&= \lim_{\epsilon \to 0}  \int_\epsilon^\infty \frac12 x^{-1/2} \, \varphi(x) \, dx \\
&= \int_0^\infty \frac12 x^{-1/2} \, \varphi(x) \, dx \\
\end{align}$$
$$\begin{align}
\langle D^2|x|_+^{1/2}, \varphi(x) \rangle
&= -\langle D|x|_+^{1/2}, \varphi'(x) \rangle \\
&= \lim_{\epsilon \to 0} \int_\epsilon^\infty \frac12 x^{-1/2} \, \varphi'(x) \, dx \\
&= \lim_{\epsilon \to 0} \left( \left[ \frac12 x^{-1/2} \, \varphi(x) \right]_\epsilon^\infty - \int_\epsilon^\infty (-\frac14) x^{-3/2} \, \varphi(x) \, dx \right) \\
&= \lim_{\epsilon \to 0} \left( -\frac12 \epsilon^{-1/2} \, \varphi(\epsilon) + \frac14 \int_\epsilon^\infty x^{-3/2} \, \varphi(x) \, dx \right) \\
\end{align}$$
$$\begin{align}
\langle D^2|x|_-^{1/2}, \varphi(x) \rangle
&= \langle |x|_-^{1/2}, \varphi''(x) \rangle \\
&= \langle |x|_+^{1/2}, \varphi''(-x) \rangle \\
&= \langle |x|_+^{1/2}, D^2\left(\varphi(-x)\right) \rangle \\
&= \langle D^2|x|_+^{1/2}, \varphi(-x) \rangle \\
&= \lim_{\epsilon \to 0} \left( -\frac12 \epsilon^{-1/2} \, \varphi(-\epsilon) + \frac14 \int_\epsilon^\infty x^{-3/2} \, \varphi(-x) \, dx \right) \\
\end{align}$$
$$\begin{align}
\langle D^2|x|^{1/2}, \varphi(x) \rangle
&= \langle D^2|x|_+^{1/2}, \varphi(x) \rangle + \langle D^2|x|_-^{1/2}, \varphi(x) \rangle \\
&= \lim_{\epsilon \to 0} \left( -\frac12 \epsilon^{-1/2} \, \varphi(\epsilon) + \frac14 \int_\epsilon^\infty x^{-3/2} \, \varphi(x) \, dx \right) \\
&+ \lim_{\epsilon \to 0} \left( -\frac12 \epsilon^{-1/2} \, \varphi(-\epsilon) + \frac14 \int_\epsilon^\infty x^{-3/2} \, \varphi(-x) \, dx \right) \\
&= \lim_{\epsilon \to 0} \left( -\frac12 \epsilon^{-1/2} \, \left(\varphi(\epsilon)+\varphi(-\epsilon)\right) + \frac14 \int_\epsilon^\infty x^{-3/2} \, \left(\varphi(x)+\varphi(-x)\right) \, dx \right) \\
&= \lim_{\epsilon \to 0} \left( - \epsilon^{-1/2} \, \varphi(0) + \frac14 \int_{|x|>\epsilon} |x|^{-3/2} \, \varphi(x) \, dx \right) \\
&= \langle \lim_{\epsilon \to 0} \left( - \epsilon^{-1/2} \, \delta(x) + \frac14 (1-\chi_{(-\epsilon,\epsilon)}(x)) |x|^{-3/2} \right), \varphi(x) \rangle \\
\end{align}$$