I would like to apply the fourier transform of the differential and I chose the following example:The heaviside function is $h(t)=1$ if $t\ge 0$ and $h(t)=0$ elsewhere. The differential of $h$ is $\delta$ the dirac function. $$F(\delta(t))= F(h'(t))=i2\pi wF(h(t))=i2\pi w({1\over 2}\delta(w)-{i\over2\pi w})=i2\pi w \delta(w)+1$$ I don't see how to interpret this result knowing already that $F(\delta(t))=1$.
ps: I used the fourier transform of the heaviside function determined via the signum function.
Note that $f(w)\delta(w)=f(0)\delta(w)$ (if $f(w)$ is continuous at $w=0$), so $w\delta(w)=0$, and, consequently, $F(\delta(t))=1$, as it should be.