Second derivative kernel density estimator (${\hat{f}}_b^{\prime \prime }(x)=\frac{1}{n{b}^3}{\sum }_i{L}^{\prime \prime }((x-X_i)/b)$)

Differentiating a twice differentiable kernel density estimator twice estimates $f^{\prime \prime }$. Its integrated variance of a kernel density estimator analogue is at most $\Vert L^{\prime \prime }{\Vert }_2^2/(n{b}^5)$. This explains why derivative estimation requires more smoothing than ordinary density estimation.