Summary: We study the expansion of derivatives along orbits of real and complex one-dimensional maps \(f\), whose Julia set \(J_f\) attracts a finite set Crit of nonflat critical points. Assuming that for each \(c\in\text{Crit}\), either \(|Df^n(f(c))|\to\infty\) (if \(f\) is real) or \(b_n\cdot|Df^n(f(c))|\to\infty\) for some summable sequence \(\{b_n\}\) (if \(f\) is complex; this is equivalent to the summability of \(|Df^n(f(c))|^{-1})\), we show that for every \(x\in J_ f\setminus \bigcup_if^{-i}(\text{Crit})\), there exist \(l(x)\leq\max_cl(c)\) and \(K'(x)>0\) such that \[ |Df^n(x)|^{l(x)}\geq K'(x)\prod_{i=0}^{s-1}D_{n_i-n_{i+1}} (c_{i+1}) \] for infinitely many \(n\). Here \(0=n_s<\dots<n_1<n_0 =n\) are so-called critical times, \(c_i\) is a point in Crit (or a repelling periodic point in the boundary of the immediate basin of a hyperbolic periodic attractor), which shadows \(\text{ orb}(x)\) for \(n_i-n_{i+1}\) iterates, and \[ D_k (c_i ) =\begin{cases} \max(\lambda,K\cdot|Df^k(f(c_i))|)&\text{if \(f\) is real},\\ \max(\lambda,K\cdot b_k\cdot|Df^k(f(c_i))|)&\text{if \(f\) is complex},\end{cases} \] for uniform constants \(K>0\) and \(\lambda>1\). If all \(c\in\text{Crit}\) have the same critical order, then \(K'(x)\) is uniformly bounded away from 0. Several corollaries are derived. In the complex case, either \(J_f=\widehat{\mathbb C}\) or \(J_f\) has zero Lebesgue measure. Also (assuming all critical points have the same order) there exist \(\kappa>0\) such that if \(n\) is the smallest integer such that \(x\) enters a certain critical neighbourhood, then \(|Df^n(x)|\geq\kappa\).