Multivariable calculus, inner products

Question

I am trying to solve this question. I have considered ith component and replaced it with $v_i/(v_i^2)^{1/2}$ and the summation form of the dot product, but cannot see how the RHS falls out, can anyone help?

Answer 1

Differentiating the ith component gives

$\displaystyle\frac{d}{dt}\left(\frac{v_i}{(v\cdot v)^{\frac{1}{2}}}\right)=\frac{(v\cdot v)^{\frac{1}{2}}\cdot v_{i}^{\prime}-v_{i}\cdot\frac{1}{2}(v\cdot v)^{-\frac{1}{2}}\cdot(2v\cdot v^{\prime})}{v\cdot v}=\frac{(v\cdot v)v_{i}^{\prime}-v_i(v\cdot v^{\prime})}{(v\cdot v)^{\frac{3}{2}}}$

${\hspace .9 in}\displaystyle=\frac{(v\cdot v)v_{i}^{\prime}-v_i(v\cdot v^{\prime})}{||v||^3}$, $\;\;\;$so the result follows from this.