= Solution
For <wall-induced motion of a planar potential dipole>, the <complex velocity> convention is $F'(z)=v_x-iv_y$. A singularity transported by the surrounding flow therefore satisfies $\dot{\bar a}=F'_{\mathrm{reg}}(a)$, where its own term $m/(z-a)$ is removed before differentiating. The images at a given instant depend on that instantaneous position, but the spatial <derivative> treats their parameters as fixed.
For the circular wall, the <derivative> of the regular image is $\bar m/(1-\bar a z)^2$. For the quadrant, differentiate its three image terms. For the sloping wall, differentiate its single image term. This gives respectively
$$
\boxed{\begin{aligned}
\text{(a)}\quad\dot{\bar a}&=\frac{\bar m}{(1-|a|^2)^2},\\
\text{(b)}\quad\dot{\bar a}&=-\frac{\bar m}{(a-\bar a)^2}+\frac{\bar m}{(a+\bar a)^2}+\frac m{4a^2},\\
\text{(c)}\quad\dot{\bar a}&=\frac{i\bar m}{(a+i\bar a)^2}.
\end{aligned}}
$$
These are equations for $d\bar z_1/dt$, as printed. Conjugating gives equations for $dz_1/dt$ if desired; no self-induced singular velocity is included.
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