MISE rate for derivative kernel density estimation ($n^{-2\beta /(2\beta +3)}$)

If $f^{\prime }\in \mathcal{N}(\beta ,L)$ and the kernel has sufficient order, the mean integrated squared error of ${\hat{f}}^{\prime }$ is bounded byBalancing the terms gives $h\asymp n^{-1/(2\beta +3)}$ and rate $n^{-2\beta /(2\beta +3)}$.