Let \(X\) be an Asplund Banach space. For a bounded subset \(A\subset X^\ast\) and \(\varepsilon>0\), we can define two derivations: \[ \begin{aligned} [A]'_\varepsilon &= \{x^\ast\in A : \text{for all }U w^\ast\text{-open half spaces containing }x^\ast, \text{ diam}(A\cap U)\geq\varepsilon\},\\ \langle A\rangle'_\varepsilon &= \{x^\ast\in A : \text{fo rall }U w^\ast\text{-neighbourhoods of }x^\ast, \text{ diam}(A\cap U)\geq\varepsilon\}, \end{aligned} \] and associated with them, two ordinal indices: \[ \begin{alignedat}{2} 2 Dz(X)_\varepsilon &= \inf\{\gamma : [B_{X^\ast}]^\gamma_\varepsilon = \emptyset\}, &\quad Dz(X) &= \sup_{\varepsilon>0}Dz(X)_\varepsilon,\\ Sz(X)_\varepsilon &= \inf\{\gamma : \langle B_{X^\ast}\rangle^\gamma_\varepsilon = \emptyset\}, &\quad Sz(X) &= \sup_{\varepsilon>0}Sz(X)_\varepsilon. \end{alignedat} \] It is clear that \(Sz(X)\leq Dz(X)\). On the other hand, G. Lancien proved that \(Sz(X)<\omega_1\) if and only if \(Dz(X)<\omega_1\), and also that there is a map \(\psi:\omega_1\longrightarrow\omega_1\) such that \(Dz(X)\leq \psi(Sz(X))\) whenever \(Sz(X)<\omega_1\). The author improves this result by showing that \(Dz(X)\leq \omega^{Sz(X)}\) for every Asplund space \(X\). The author studies this kind of derivation processes with respect to a general measure of non-compactness \(\eta\) playing the role of diam above. Examples of such measures are \(\eta=\beta_p\), \(p\in [1,\infty]\), related to superreflexivity [cf. \textit{G. Pisier}, Isr. J. Math. 20, 326--350 (1975; Zbl 0344.46030)], the so called Kuratowski measure [considered in \textit{F. García, L. Oncina, J. Orihuela} and \textit{S. Troyanski}, J. Convex Anal. 11, No. 2, 477--494 (2004; Zbl 1067.46002)] and measures of noncompactness obtained from others by iteration. He obtains applications of this to locally uniformly convex renorming of Banach spaces, obtaining new proofs and improvements of results of Troyanski and Moltó-Orihuela-Troyanski.