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:

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

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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