estimate on the norm of Aexp(tA)

Question

Let A be a infinitesimal generator of an analytic semigroup $\{S(t)\}_{t}$ of contraction operators on $X$ that is a Banach space. I know that $S(t)X \subset D(A)$ for $t>0$. How can I prove from this and from the spectral theorem that for $t>0$ $$||AS(t)|| \le \dfrac{C}{t}?$$