  We give a slight refinement to the process by which estimates for exponential
sums are extracted from bounds for Vinogradov's mean value. Coupling this with
the recent works of Wooley, and of Bourgain, Demeter and Guth, providing
optimal bounds for the Vinogradov mean value, we produce a powerful new $k$-th
derivative estimate. Roughly speaking, this improves the van der Corput
estimate for $k\ge 4$. Various corollaries are given, showing for example that
$\zeta(\sigma+it)\ll_{\varepsilon}t^{(1-\sigma)^{3/2}/2+\varepsilon}$ for $t\ge
2$ and $0\le\sigma\le 1$, for any fixed $\varepsilon>0$.
