  We determine a connection between the weight of a Boolean function and the
total weight of its first-order derivatives. The relationship established is
used to study some cryptographic properties of Boolean functions. We establish
a characterization of APN permutations in terms of the weight of the
first-order derivatives of their components. We also characterize APN functions
by the total weight of the second-order derivatives of their components. The
total weight of the first-order and second-order derivatives for functions such
as permutations, bent, partially-bent, quadratic, plateaued and balanced
functions is determined.
