I'm sorry if this is supposed to be something basic but I'm not being able to understand if r is as given above, how have they worked out r dot? What have they differentiated the x,y and z coordinates with respect to? r dot means r differentiated with respect to what?
Any help would be much appreciated.
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It seems that $\underline {r}$ is a vector in the $x-y$ plane expressed in polar coordinates with $r$ and $\theta$ depending on a same parameter $t$ (it can be the time). The notation $\dot r$ and $\dot \theta$ are used to indicate the derivative with respect the parameter $t$ and the chain rule is applied to the components of the vector.
Concentrate first on the $x$ parameterisation. Write $$x(r(t), \theta(t)) = r(t) \cos (\theta(t))$$
Remember that the product rule for differentiation is $$\frac{d}{dt}(uv) = u\frac{dv}{dt}+v\frac{du}{dt}$$ Hence \begin{align} \frac{dx}{dt} &= \dot{x} \\ &= \frac{d}{dt}(r \cos \theta ) \\ &= \frac{dr}{dt}\cos \theta + r \frac{d}{dt}\cos \theta \\ &= \frac{dr}{dt}\cos \theta - r \sin \theta \frac{d \theta}{dt} \\ &= \dot{r}\cos \theta - r \dot{\theta} \sin \theta \end{align} And the remaining two derivatives can be found similarly.