If two $n$-dimensional vectors $\mathbf u$ and $\mathbf v$ are functions of time, the derivative of their dot product is given by $\frac{\mathrm d}{\mathrm dt}(\mathbf u\cdot\mathbf v) = \mathbf u\cdot\frac{\mathrm d\mathbf v}{\mathrm dt} + \mathbf v\cdot\frac{\mathrm d\mathbf u}{\mathrm dt}$ This is analogous to (and indeed, is easily derived from) the product rule for scalars, $\frac{\mathrm d}{\mathrm dt}(ab) = a\frac{\mathrm db}{\mathrm dt} + b\frac{\mathrm da}{\mathrm dt}$.
Therefore, $\frac{\mathrm d}{\mathrm dt} \lVert\mathbf v\rVert^2 = \frac{\mathrm d}{\mathrm dt}(\mathbf v\cdot\mathbf v) = \mathbf v\cdot\frac{\mathrm d\mathbf v}{\mathrm dt} + \mathbf v\cdot\frac{\mathrm d\mathbf v}{\mathrm dt} = 2\mathbf v\cdot\frac{\mathrm d\mathbf v}{\mathrm dt}$ just like $\frac{\mathrm d}{\mathrm dt} a^2 = 2a\frac{\mathrm da}{\mathrm dt}$. Halve that, and you have the result you need.