Derivatives in Multivariable Calculus

The derivative of $f(x,y)$ in the direction $\vec{v}=(v_1,v_2)$, at point $p$, is:
$$(f_x(p),f_y(p)) \cdot (v_1,v_2) = v_1 f_x(p) + v_2 f_y(p) $$

But wait: $f_x(p)$ and $f_y(p)$ measure how $f$ changes in directions of $x$ and $y$. Then the formula above claims that knowing only this piece of information, we can know how $f$ changes in any other direction as well?! How could this be true? I will answer this, and end with a eulogy of derivatives in general!