How to reconcile two definitions of the gradient (partial derivative vs Frechet derivative)?

Question

There are two definitions of the gradient of a function $f$ at $x$

1. Stacked partial derivatives of $f$ $$ \nabla f(x) = \left(\frac{\partial f}{\partial x_1}, ..., \frac{\partial f}{\partial x_n}\right)(x)$$

2. The Frechet derivative

$$ \lim_{h \to 0} \frac{f(x+h) - f(x) - v^Th} {\|h\|} = 0 $$ The unique vector $v$ is the gradient of $f$ at $x$.

How can I reconcile between these two definitions? It is not obvious to me that this vector $v$ must be the stacked partial derivatives of $f$.

Can someone help?

Answer 1

In the second definition if you take $h=(h_1,0,..,0)$ and let $h_1 \to 0$ you see immediately that $v_1=\frac {\partial f} {\partial x_1}$. Similarly, $v_i=\frac {\partial f} {\partial x_i}$ for all $i$.

However, the existence of partial derivatives does not guarantee the existence of Frechet derivative. For counter-examples see Is diffirentiability in finite dimensional space is equivalent to the existence of partial derivatives