Found a strange differential equation with quaternions and $\exp()$, how do I solve?

This looks like directional derivatives in some kind of linear-algebra formalism. $\delta_q$ would then be the direction of derivation or the operator that extracts the tangent direction from a point in the tangent space (as fibre bundle) over $q$.

In another formalism one could write $$ \frac{d}{dt}q^n = \sum_{k=1}^nq^{n-k}\dot qq^{k-1} $$ where $q(t)$ is some differentiable path, $q=q(0)$, $\dot q=\dot q(0)=\frac{d}{dt}q(t)|_{t=0}$ etc.

Now we want to know if in the derivative of the exponential $e^{q(t)}$ one can factor out the linearly occurring $\dot q$. This is usually done working not from the series of the exponential, but from its power limit definition $$ e^q=\lim_{n\to\infty}e_n(q)=\lim_{n\to\infty}\left(1+\frac qn\right)^n $$ so that $$ \frac{d}{dt}e_n(q)=\frac1n\sum_{k=1}^ne_n(q)^{1-k/n}\,\dot q \,e_n(q)^{k/n-1/n}. $$ In the limit, claiming that everything is tame and exists, and reading the sum as a Riemann sum, thus replacing that with the corresponding integral, $$ \frac{d}{dt}e^q=\int_0^1e^{(1-s)q}\dot qe^{sq}\,ds. $$ This does not look very amenable to factoring $\dot q$ out. In matrices and Lie groups one continues with the adjoint operator and its exponential I do not see off-hand how that translates to quaternions in some organic way.