Determine the differential of the regular distribution $T_f$ in the space $D'$(continuous dual of $D$) for $f(x)=H(x)cos(x)$, where $H$ is a Heaviside function and $x\in \Bbb{R}$.
Since $H(x) = +1$ for $x>0$, and $H(x) = 0$ for $x\leq 0,$ It can be represented as a distribution: $$\langle T_f,\varphi\rangle=\int f(x)\varphi(x)=\int_{0}^{\infty}\cos x\varphi(x)$$ to allow test functions to be any function $\varphi(x)$ that is infinitely smooth.
This distribution has a distributional derivative:
$$\langle T_f',\varphi\rangle=-\int f(x)\varphi'(x)=\int_{0}^{\infty}\cos x\varphi'(x)=\varphi(0)-\int_{0}^{\infty}\sin x\varphi'(x)$$, using integration by parts as $\varphi(\infty)=0$ because $\varphi$ has compact support zero outside the bounded set. Am I correct? Thanks for your comment.