Distributional derivative of a continuous nonnegative function

Question

Let $H$ be a Hilbert space and $f: [0,+\infty)\to H$ be a $C^1$ function such that $\lim_{t\to+\infty}f(t)=0$. Does $\|f(t)\|$ have a distributional derivative? If yes, is it true to say that "if for some $t_0>0$, $\frac{d}{dt}\|f(t)\|<0$, then $\|f(t)\|$ is decreasing on a neghberhood of $t_0$"?