Equivalent notions of the differential on a manifold. Why are they equivalent?

Question

I would like to know how one sees the equality between these two notions of the differential:

$\mathbf{v}_p[f]=\frac{d}{dt}(f(\mathbf{p}+t\mathbf{v})|_{t=0}$

and

$df(\mathbf{v})=\sum_j{v^j(p)}\frac{\partial f}{\partial x^j}(p)$.

Here $df:M^n_p \to \mathbb{R}$.

Answer 1

It follows from the chain rule,

$$ \frac{d}{dt} f(c(t)) = \nabla f(c(t)) \cdot c'(t)$$

In your problem $c(t) = p+tv$.