# Dual vector

$$f(\vec{x}+\epsilon \vec{y})-f(\vec{x})=\epsilon \mbox{d}f_{\vec{x}}(\vec{y})+O(\epsilon^2)$$.
Is ##\mbox{d}f_{\vec{x}}(\vec{y})## dual vector and why? Is it because ##\mbox{d}## is linear transformation? Also why equality
$$f(\vec{x}+\epsilon \vec{y})-f(\vec{x})=\epsilon \mbox{d}f_{\vec{x}}(\vec{y})+O(\epsilon^2)$$
is correct?