  The derivatives of Nash functions are Nash functions which are derived
algebraically from their minimal polynomial equations. In this paper we show
that, for any non-Nash analytic function, it is impossible to derive its
derivatives algebraically, i.e., by using linearity and Leibniz rule finite
times. In fact we prove the impossibility of such kind of algebraic
computations, algebraically by using K{\" a}hler differentials. Then the notion
of Leibniz complexity of a Nash function is introduced in this paper, as a
computational complexity on its derivative, by the minimal number of usages of
Leibniz rules to compute the total differential algebraically. We provide
general observations and upper estimates on Leibniz complexity of Nash
functions, related to the binary expansions, the addition chain complexity, the
non-scalar complexity and the complexity of Nash functions in the sense of
Ramanakoraisina.
