Multivariable Taylor expansion ($f(a+h)=f(a)+Df(a)h+\frac{1}{2}{h}^T{H}_f(a)h+O(\Vert h{\Vert }^3)$)

For a scalar function with continuous third derivatives near $a$, apply the one-dimensional Taylor theorem to $g(t)=f(a+th)$. The chain rule gives $g^{\prime }(0)=Df(a)h$ and $g^{\prime \prime }(0)=h^T{H}_f(a)h$, proving the displayed formula. Here $H_f$ is the Hessian matrix. The finite polynomial part is the Multivariate Taylor polynomial; the remainder controls its accuracy.

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 $s^2$.