In [Abh. Math. Semin. Univ. Hamb. 16, 1-28 (1949; Zbl 0035.06004)], \textit{G. Bol} proved: Suppose \(r\in \mathbb{Z}\), \(r\geq 0\); then \[ D^{(r+1)} \Biggl\{(c\tau+d)^r F\biggl( \frac{a\tau+b}{c\tau+d} \biggr)\Biggr\}= (c\tau+d)^{-r-2} F^{(r+1)} \biggl( \frac{a\tau+b}{c\tau+d} \biggr), \] for \(ad-bc=1\) and any \(F\) defined on the complex plane \(\mathbb{C}\) with sufficiently many derivatives. While this differentiation formula is easy to prove (e.g., by induction on \(r\)), it has (at least) two important consequences: (i) If \(F\) is an automorphic form (AF) of weight \(-r\) \((r\in \mathbb{Z}\), \(r\geq 0)\), then \(F^{(r+1)}\) is an AF of weight \(r+2\), with respect to the same group and multiplier system (MS). (ii) If \(f\) is an AF of weight \(r+2\) \((r\in \mathbb{Z}\), \(r\geq 0)\), and \(f= D^{(r+1)}F\), then \(F\) is an Eichler integral of weight \(-r\), with respect to the same group and MS. (This means that \(F\) has transformation properties analogous to those of an AF of weight \(-r\), but modified by the presence of Eichler periods, polynomials in \(\tau\) of degree \(\leq r\).) This second consequence underlies the formation of the Eichler cohomology groups associated with AF's. The article under review generalizes these ideas to the technically more complicated situation in which AF's are replaced by Jacobi forms, functions of two complex variables with properties abstracted from those of Jacobi's classic theta-functions. A Jacobi form has two associated real parameters, the weight (similar to the weight of an AF, but necessarily half-integral) and the index, an integer. (The index has its origins in the doubly-periodic behavior of the theta-functions.) In the author's generalization of (1), \(F\) is replaced by a function \(f\) on \(H\times \mathbb{C}\) (\(H\) is the upper half-plane), and differentiation by the heat operator \[ L_{(m)}= \frac{2m}{\pi i} \frac{\partial}{\partial\tau}- \frac{1}{(2\pi i)^2} \frac{\partial^2}{\partial z^2}. \] (Here, \(\tau\in H\) and \(z\in \mathbb{C}\).) This generalization of (1) (Theorem 4.1) leads to a potentially important generalization of property (i), from AF's to Jacobi forms: If \(f\) is a Jacobi form of weight \(-r+\frac{1}{2}\) and index \(m\) \((r,m\in \mathbb{Z}\), \(r\geq 0)\), then \(L_{(m)}^{r+1}(f)\) is a Jacobi form of weight \(r+ \frac{5}{2}\) and index \(m\). The paper closes with an interesting discussion, inspired by property (ii) of AF's, on connections with periods of modular forms.