Square root of the second derivative operator

Question

Let $p>1$ and $X=L^p(a,b)$ be the space of $L^p$-integrable functions on $(a,b)$ and consider the operator $A f = f''$ with domain $D(A)=W^{2,p}(a,b)\cap W^{1,p}_0(a,b)$. Now, my question is : can we say that the square root $(-A)^\frac{1}{2}$ coincides with the first derivative on $(a,b)$?