TITLE: $\lim_{h \rightarrow 0} \frac{f(x+h \hat{u})-f(x)}{h}=\frac{\partial f}{\partial x}u_x+ \cdot \cdot \cdot$
QUESTION [1 upvotes]: How are the following directional derivative definitions equivalent: $$\lim_{h \rightarrow 0} \frac{f(x+h \hat{u})-f(x)}{h}=\frac{\partial f}{\partial x}u_x+\frac{\partial f}{\partial y}u_y+\frac{\partial f}{\partial z}u_z+ \cdot \cdot \cdot$$
where $u_i$ are unit vectors.
as given here:
http://mathworld.wolfram.com/DirectionalDerivative.html

REPLY [1 votes]: That is not a definition ! If $f$ is differentiable at $x$, then the directional derivative is given by
$\lim_{h \rightarrow 0} \frac{f(x+h \hat{u})-f(x)}{h}=\frac{\partial f}{\partial x}(x)u_x+ \frac{\partial f}{\partial y}(x)u_y+\frac{\partial f}{\partial z}(x)u_z$
where $\hat{u}=(u_x,u_y,u_z)$
{"set_name": "stack_exchange", "score": 1, "question_id": 2007683}