I would like to show that the finite part of $\frac{1}{x_+}=\mathbb{1}_{[0,+\infty[}\frac{1}{x}$, noted Fp$\left(\frac{1}{x_+}\right)$ and given by $$\left\langle\text{Fp}\left(\frac{1}{x_+}\right),\varphi\right\rangle=\lim_{\varepsilon\to 0^+}\int_\varepsilon^\infty\frac{\varphi(x)}{x}dx + \varphi(0)\log(\varepsilon)$$ is a tempered distribution.
I have already shown that the limit exists and its value is $$\lim_{\varepsilon\to 0^+}\int_\varepsilon^\infty\frac{\varphi(x)}{x}dx + \varphi(0)\log(\varepsilon)=-\int_0^\infty\log(x)\varphi'(x)dx$$ but I don't know how to prove that $C\sup\limits_{x\in\mathbb{R}}|\langle x \rangle^{2}\varphi'(x)|$ is an upper bound of its absolute value, for some $C>0$.
Could anyone help me?
Thanks in advance.