Relation between gradient vector and partial derivatives

Statement

Suppose $f$ is a real-valued function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . Suppose $(a_{1}, a_{2}, \dots, a_{n})$ is a point in the domain of $f$ such that the gradient vector of $f$ at $(a_{1}, a_{2}, \dots, a_{n})$ , denoted $(\nabla f) (a_{1}, a_{2}, \dots, a_{n})$ , exists. Then, the partial derivatives of $f$ with respect to all variables exist, and the coordinates of the gradient vector are the partial derivatives. In other words:

$(\nabla f) (a_{1}, a_{2}, \dots, a_{n}) = ⟨ f_{x_{1}} (a_{1}, a_{2}, \dots, a_{n}), f_{x_{2}} (a_{1}, a_{2}, \dots, a_{n}), \dots f_{x_{n}} (a_{1}, a_{2}, \dots, a_{n}) ⟩$

Related facts

Existence of partial derivatives not implies differentiable: It is possible that all the partial derivatives exist at a point but the gradient vector doesn't.
Relation between gradient vector and directional derivatives