 Directional error of a first-order Taylor approximation

For a fixed unit direction $v$, the multivariable Taylor expansion gives the displayed leading error. Equal directional derivatives in two directions do not imply equal approximation errors: the linear changes depend on the gradient, while the quadratic errors depend on the Hessian matrix. For example, $f(x,y)=x+y+2x^2+y^2$ at the origin has equal derivatives in the two coordinate directions but errors $2s^2$ and $s^2$.