Take a function \(f\). Define the sequence \(f_i\) by \[ \begin{align*} D_0(f)(x) &= f(x)\\ D_{n+1}(f)(x) &= f'(D_n(f)(x)) \end{align*} \] Then consider the limit \(D_\infty(f)\) \[ \mathrm{lim}_{n\rightarrow\infty}D_n(f) \] Now for any polynomial \(p(x)\), clearly \(D_\infty(p)=0\). On the other hand, for \(f(x)=Ae^{Bx}\), we have \[\begin{align*} D_\infty(Ae^{Bx})&\rightarrow\infty\textrm{ if \(B>1\)}\\ D_\infty(Ae^x)&=Ae^x\\ D_\infty(Ae^{Bx})&=0\textrm{ if \(0\leq B<1\)} \end{align*}\] For \(sin(x)\) this limit does not exist.