  We give an explicit version for van der Corput's $d$-th derivative estimate
of exponential sums.
  $ \textbf{Theorem.}$ Let $X$, and $Y\in\mathbb{R}$ be such that $\lfloor
Y\rfloor>d$ where $d\ge3$ is a natural number. Let
$f\colon(X,X+Y]\to\mathbb{R}$ be a real function with continuous derivatives up
to the order $d$. Assume that $0<\lambda\le f^{(d)}(x)\le\Lambda$ for $X<x\le
X+Y$. Denote by $D=2^d$. Then \begin{equation}\Bigl|\frac{1}{Y}\sum_{X<n\le
X+Y}e(f(n))\Bigr|\le\max\Bigl\{A_d\Bigl(\frac{\Lambda}{\lambda Y}\Bigr)^{2/D},
B_d\Bigl(\frac{\Lambda^2}{\lambda}\Bigr)^{1/(D-2)},C_d(\lambda
Y^d)^{-2/D}\Bigr\},\end{equation} where $A_d$, $B_d$, and $C_d$ are explicit
constants. They depend on $d$ but for $d\ge2$ for example $A_d< 7.5$, $B_d<5.8$
and $C_d<10.9$.
  We follow the reasoning of van der Corput in three papers published in 1937,
that contained an error. I correct this error and try to get the smallest
possible constants. We apply this theorem to zeta sums, giving the best choice
of $d$ in each case. Also, we prove that our Theorem implies Titchmarsh's
Theorem 5.13.
