i take a course about partial differential equations and we follow the book by Rauch.
In the appendix of the book are some excercises about computing distributional derivation and i´m struggling about to compute
$\frac{\mathrm{d^{k}}}{\mathrm{dx^{k}}} |\sin(x)| $.
My first idea was to start with the formula
$ \langle\frac{\mathrm{d^{k}}}{\mathrm{dx^{k}}} |\sin(x)| , \varphi \rangle = (-1)^{k} \langle|\sin(x)|,\frac{\mathrm{d^{k}}}{\mathrm{dx^{k}}} \varphi \rangle = \int_{-\infty}^{\infty}|\sin(x)|\frac{\mathrm{d^{k}}}{\mathrm{dx^{k}}} \varphi \mathrm{dx} $
but then i got no idea how to go on, because i dont how to split the integration interval that i can use partial integration.
I hope someone can help me with this "easy" calculation