How is the Chain Rule Applied in Geometric Tangent Vectors?

[BrainHurts]

So let $ℝ^{n}_{a}$={(a,v) : a $\in$ $ℝ^{n}$, v $\in$ $ℝ^{n}$}

so any geometric tangent vector, which is an element of $ℝ^{n}_{a}$ yields a map

D_v|af = D_vf(a) = $\frac{d}{dt}|_{t=0}$f(a+tv)

this operation is linear over ℝ and satisfies the product rule

D_v|a(fg) = f(a)D_vg + g(a)D_vf

if v|a = $\sum_{i=1}^n$ v_ie_i|a, then by the chain rule
D_v|af can be written as:

D_v|af $\sum_{i=1}^n$ v_i $\frac{∂f}{∂x_{i}}(a)$

not seeing how the chain rule applies and how the result as such.

 

[dx]

x_i(t) = a_i + tv_i

(d/dt) f(a + tv) = (d/dt) f(x_i(t))

= Σ (∂f/∂x_i) (dx_i/dt)

= Σ v_i(∂f/∂x_i)