Derivatives of polar coordinates

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I've got a problem for which I'm trying to calculate $\ddot r$.

The problem is right here for the sake of reference.http://i.imgur.com/5tYaRv7.png


So far, I've got that: $$\ddot r=\frac{d}{dt}v_r=\frac{dv_r}{dr}\frac{dr}{dt}$$

And since: $$\dot r= \frac{dr}{dt}$$

I simplified down to: $$\ddot r=\dot r \frac{dv_r}{dr}$$

Now I'm stuck, because I want to assume $v_r$ is a constant, in which case the whole $\ddot r$ term goes to zero. When I do this I get the wrong answer, however.

How could I take the derivative of $v_r$ with respect to $r$ in this situation?

Thanks in advance for any and all help!