I understand there are multiple ways of of proving the product rule for the derivative of an inner product, though I cannot figure out how to do this one specifically:
let $\alpha,\beta :R \rightarrow R^n$ be differentiable functions. If $ f(t)=\langle \alpha(t),\beta(t) \rangle,$ using only these three rules (IE the conditions for inner product):
$\langle x,x \rangle > 0$ if $x \not= 0$
$\langle x,y \rangle = \langle y,x \rangle $
$ \langle ax+by,z \rangle = a\langle x,z \rangle + b\langle y,z \rangle $
$f'(t) = \langle \alpha(t), \beta'(t) \rangle + \langle \alpha'(t), \beta(t) \rangle $
Basically I understand how to do it using summations, the normal product rule and and the linearity of summations but I do not understand how to get this result another way. There was another question asked that did it by taking the limits, which also makes sense to me.
thanks for any suggestions/help.