This comes as a follow-up to other question. The original was about discretizing a nonlinear curvilinear dynamics equation, this follow up is about taking the Jacobian and simplifying such as remove division by the turn rate $\omega$, which is expected to be small.
Take the discretized $x$ dynamics $$ \begin{aligned} x\vert_{t+T} &= x\rvert_t +(v\rvert_t T+\tfrac12aT^2)\cos(\psi\vert_t+\tfrac12\omega T)\operatorname{sinc}(\tfrac12\omega T) + \gamma \sin(\psi\rvert_t+\tfrac12\omega T) \end{aligned} $$ where the true value and approximations for $\gamma$ are $$ \begin{aligned} \gamma_{x} &= \frac{aT}{\omega} \left[ \cos(\tfrac12\omega T) - \operatorname{sinc}(\tfrac12\omega T) \right]\\ &\approx -\tfrac1{12}a\omega T^{3} \cos(\tfrac{\omega T}{\sqrt{20}})\\ &\approx -\tfrac1{12}a\omega T^{3} \operatorname{sinc}(\omega T \sqrt{\tfrac{3}{20}})\\ \end{aligned} $$
Taking the derivative of this, using the true value for $\gamma$, with respect to $\omega$, in MATLAB for example, leads to something like $$ \frac{\cos\left(\psi +\frac{\mathrm{dt}\,\omega }{2}\right)\,\cos\left(\frac{\mathrm{dt}\,\omega }{2}\right)\,\left(\frac{a\,{\mathrm{dt}}^2}{2}+v\,\mathrm{dt}\right)}{\omega }-\frac{\sin\left(\psi +\frac{\mathrm{dt}\,\omega }{2}\right)\,\sin\left(\frac{\mathrm{dt}\,\omega }{2}\right)\,\left(\frac{a\,{\mathrm{dt}}^2}{2}+v\,\mathrm{dt}\right)}{\omega }-\frac{2\,\cos\left(\psi +\frac{\mathrm{dt}\,\omega }{2}\right)\,\sin\left(\frac{\mathrm{dt}\,\omega }{2}\right)\,\left(\frac{a\,{\mathrm{dt}}^2}{2}+v\,\mathrm{dt}\right)}{\mathrm{dt}\,\omega ^2}+\frac{a\,{\mathrm{dt}}^2\,\cos\left(\psi +\frac{\mathrm{dt}\,\omega }{2}\right)\,\left(\cos\left(\frac{\mathrm{dt}\,\omega }{2}\right)-\frac{2\,\sin\left(\frac{\mathrm{dt}\,\omega }{2}\right)}{\mathrm{dt}\,\omega }\right)}{2\,\omega }-\frac{a\,\mathrm{dt}\,\sin\left(\psi +\frac{\mathrm{dt}\,\omega }{2}\right)\,\left(\frac{\mathrm{dt}\,\sin\left(\frac{\mathrm{dt}\,\omega }{2}\right)}{2}+\frac{\cos\left(\frac{\mathrm{dt}\,\omega }{2}\right)}{\omega }-\frac{2\,\sin\left(\frac{\mathrm{dt}\,\omega }{2}\right)}{\mathrm{dt}\,\omega ^2}\right)}{\omega }-\frac{a\,\mathrm{dt}\,\sin\left(\psi +\frac{\mathrm{dt}\,\omega }{2}\right)\,\left(\cos\left(\frac{\mathrm{dt}\,\omega }{2}\right)-\frac{2\,\sin\left(\frac{\mathrm{dt}\,\omega }{2}\right)}{\mathrm{dt}\,\omega }\right)}{\omega ^2} $$ and using an inbuilt simplification tool leads to $$ \frac{\mathrm{dt}\,\cos\left(\psi +\mathrm{dt}\,\omega \right)\,\left(v+a\,\mathrm{dt}\right)}{\omega }-\frac{2\,a\,\left(\cos\left(\psi +\mathrm{dt}\,\omega \right)-\cos\left(\psi \right)\right)}{\omega ^3} - \frac{v\,\sin\left(\psi +\mathrm{dt}\,\omega \right)-v\,\sin\left(\psi \right)+2\,a\,\mathrm{dt}\,\sin\left(\psi +\mathrm{dt}\,\omega \right)}{\omega ^2} $$ Is there any systematic way to approach the simplification of this under the assumption that $\vert\omega\vert$ is small?