# Confusion regarding use of differentiation and unit vectors

Hey, everyone. I am going to post a question—but it's not the question I need help with. It's something deeper (and way more troubling).

Consider a particle of mass m subject to an isotropic two-dimensional harmonic central force F= −k$\vec{r}$, where k is a positive constant. At t=0, we throw the particle from position $\vec{r}_0$ = A$\hat{x}$ with velocity $\vec{v}_0$ = V$\hat{y}$. Show that the trajectory of the particle is, in general, an ellipse.

So my plan is just to solve the equation for Newton's 2nd law of motion, get a 2nd order differential equation, etc. The confusion comes in when I ask myself "When do I need to make use of the chain rule?"

It seems easy enough to me to just take the equation $k\vec{r} + m\ddot{\vec{r}} = 0$ at face value but (and this may seem like a silly question)...

... isn't $\vec{r} = r\hat{r}$?
So $\dot{\vec{r}} = \dot{r}\hat{r} + r\dot{\hat{r}}$?

... and then I use the chain rule again to get the second derivative: $\hat{r}\ddot{r} + \dot{r}\dot{\hat{r}} + \dot{r}\dot{\hat{r}} + \ddot{\hat{r}}r$

I'm sure I'm over-complicating things, but this is the type of confusion I always end up wasting tons of my time on, and I have a real hard time finding an answer in textbooks or Wikipedia.

The only way I can consolidate the two methods is by thinking: The force isn't dependent on θ, so the 1st and 2nd derivatives of θ are zero. Since the 1st and 2nd derivatives of $\hat{r}$ depend on the derivatives on theta, those are also zero—and that ugly combination above simplifies to $\ddot{r}\hat{r}$.

Is this correct? It sounds fine to me, but questions like this (can I do this? Or should I approach it, mathematically, like this? etc.) take up so much of my time that I end up spending multiple hours on problems I realistically should spend only a couple of minutes on.