<s>[INST] Transform $dx/dt$ to $dr/dt$ polar coordinates I've had to screenshot the question and post it as a photo

 [/INST] The transformations can be written as
$$\tag 1  r^2 = x^2 + y^2~~ \text{and}~~ \theta = \tan^{-1}\left(\frac{y}{x}\right)$$
Differentiating $(1)$
$$r^2 = x^2 + y^2 \implies 2 r r' = 2x x' + 2y y' \implies r ' = \dfrac{x x' + y y'}{r}$$
Substituting $x = r \cos \theta, y = r \sin \theta$, we have
$$\begin{align} r' &= \dfrac{x x' + y y'}{r} \\&=  \dfrac{(r \cos \theta \left(-r \cos \theta \left(r^2 \right)-r \sin \theta +r \cos \theta\right)+r \sin \theta \left(-r \sin \theta \left(r^2 \right)+r \sin \theta+r \cos \theta)\right)}{r}\\ &= r(1-r^2)\end{align}$$
We can also derive
$$\theta = \tan^{-1}\left(\frac{y}{x}\right) \implies \theta' = \dfrac{xy'-yx'}{r^2}$$
Make the appropriate substitutions, do some trig simplifications and cleanup to derive  $\theta'$.
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