This paper provides instability criteria for linear dynamic equations on time scales. More detailed, linear systems of the form \[ x^\Delta=A(t)x \] are considered, where \(A:{\mathbb T}\to{\mathbb R}^{n\times n}\) is an \(rd\)-continuously differentiable function. Special cases of this class include ODEs and difference equations. In the time-invariant case \(A(t)\equiv A\) it is well known that stability properties depend on the spectrum of \(A\) (cf.~the reviewer, \textit{S. Siegmund} and \textit{F. Wirth} [Discrete Contin. Dyn. Syst. 9, No. 5, 1223--1241 (2003; Zbl 1054.34086)] for a general time scale approach). The present paper develops eigenvalue criteria under which a ``slowly time varying'' linear dynamic equation is unstable. ``Slowly varying'' means that the derivative \(A^\Delta\) has to have a small global bound. The corresponding tools include a quadratic Lyapunov function. As usual in stability theory for dynamic equations, the time scale is assumed to have bounded graininess \(\mu\); beyond that, however, also \(\mu^\Delta\) is supposed to exist as a bounded function. Remark: Related results for linear dynamic equations having an exponential dichotomy can be found in [the reviewer, J. Math. Anal. Appl. 289, No. 1, 317--335 (2004, Zbl 1046.34076)].