# Dual vector

## Main Question or Discussion Point

$$f(\vec{x}+\epsilon \vec{y})-f(\vec{x})=\epsilon \mbox{d}f_{\vec{x}}(\vec{y})+O(\epsilon^2)$$.
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?

The equality is correct because that is how $df$ is defined. And it is a dual vector because it takes vectors as arguments and gives a number as a result, and it is linear in the argument (the $\vec{y}$ in you expression).