Intuition for property of derivations Revisiting Lee over the summer and proved a lemma, but I'm blanking on the intuition for why it should be true: Given $f,g \in C^ \infty$ and $\omega \in T_p\mathbb{R}^ n$, then if $f(p) = g(p) = 0, \ \omega (fg) = 0$. The proof is one line, but I'm not sure why if two functions vanish at a point, then all directional derivatives at that point of the product function should be $0$. In particular, if $f$ vanishes at a point, then the directional derivatives of $f^ 2$ are all $0$. Or is my interpretation of $\omega$ as a directional derivative not accurate? 

Think about smooth functions vanishing at the origin in $\Bbb R^2$. Such functions
look like
$$f(x,y)=ax+by+\text{quadratic and higher terms}$$
and
$$g(x,y)=a'x+b'y+\text{quadratic and higher terms}.$$
Then
$$f(x,y)g(x,y)=aa'x^2+(ab'+a'b)xy+bb'y^2+\text{cubic and higher terms}$$
so any directional derivative of $fg$ at the origin vanishes.
