Exercise 1

\[ \frac{\mathrm{d}v^2}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} (v \cdot v ) \]

We use the product rule and this yields: \[ = \frac{\mathrm{d}}{\mathrm{d}t} v \cdot v + v \cdot \frac{\mathrm{d}}{\mathrm{d}t} v \] \[ = 2 \frac{\mathrm{d}}{\mathrm{d}t} v \cdot v = 2 v \dot{v} \]