Derivative of Dot Product of Vector-Valued Functions/Proof 1

Proof
Let:
 * $\mathbf a: x \mapsto \tuple {\map {a_1} x, \map {a_2} x, \ldots, \map {a_n} x}$
 * $\mathbf b: x \mapsto \tuple {\map {b_1} x, \map {b_2} x, \ldots, \map {b_n} x}$

Then: