= Solution
Differentiate the <polar form of a complex number> using the <product rule> and <chain rule>:
$$
\boxed{\dot z=(\dot r+ir\dot\theta)e^{i\theta}.}
$$
A second <differentiation> gives
$$
\begin{aligned}
\ddot z
&=\bigl[\ddot r+i(\dot r\dot\theta+r\ddot\theta)
+i\dot\theta(\dot r+ir\dot\theta)\bigr]e^{i\theta}\\
&=\boxed{\bigl[(\ddot r-r\dot\theta^2)
+i(2\dot r\dot\theta+r\ddot\theta)\bigr]e^{i\theta}.}
\end{aligned}
$$
The two contributions to $2\dot r\dot\theta$ arise separately from differentiating $r\dot\theta$ and from differentiating the moving exponential factor.
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