Distributional derivative of $\delta'*1$ is equal to $(\delta*1)'=1'=0$ when it acts on any $f\in C_c^\infty(\mathbb{R})$. However, what happens if $f\in C^\infty(\mathbb{R})$ does not have a compact support?
In particular, when proving the non-associativity of convolution of distributions $$(H*\delta')*1=\delta*1=1\ne0=H*0=H*(\delta'*1)$$ where $H$ is the Heaviside function whose distributional derivative is the delta function, we have used the property $\delta'*1=0$ in distribution. However, if we consider how $H*(\delta'*1)$ acts on $f\in C_c^\infty(\mathbb{R})$. It is equivalent to $\delta'*1$ acting on the convolution of $f(x)$ with $H(-x)$. This convolution as a function no longer has bounded support. How do we define the action of $\delta'*1$ on it?