I am reading Introduction to Smooth Manifolds by Lee and on page 52 about Tangent Vectors, the author defines the map
$$ D_{v|_a}: C^{\infty}(\mathbb{R}^n) \rightarrow \mathbb{R} $$ which takes the directional derivative in the direction $v$ at $a$:
$$ D_{v|_a} f = D_v f(a) = \frac{d}{dt} \bigg\vert_{t=0} f(a + tv). $$ He then proceeds to say that if $$ v_a = v^i e_i|_a (\text{Einstein notation}) $$ in terms of the standard basis then by the chain rule, $$ D_{v|_a}f = v^i \frac{\partial f}{\partial x^i}(a). $$ But I don't understand how can the chain rule get us to the final expression. Any help is appreciated.
By definition (3.1) on this page, $$ D_v|_a f = \frac{d}{dt}|_{t = 0} f(a + vt) = \frac{d}{dt}|_{t = 0} f(a + v^i e_i t). $$ To show this last expression is equal to the final expression you've written, you use the chain rule. For example, if $v$ was just $e_1$, for instance, $$ \frac{d}{dt}|_{t = 0} f(\underbrace{a + e_1 t}_{(g^1(t), \ldots, g^n(t))}) = \frac{\partial f}{\partial x^i}(a) \cdot \frac{dg^i}{dt}(t = 0) = \frac{\partial f}{\partial x^1}(a)\cdot 1. $$