Assuming we start with the derivations, $w_p(f)$ that are linear and satisfy the product rule.
Is it possible to show that what $w(f) = D_v(f)$ without first defining what a directional derivative, Taylor's theorem, etc.
In other words, can one show that $w(f) = \lim_{t \rightarrow 0} \frac{f(p + vt) - f(p)}{t}$ for some $v$ without first defining derivatives.