= Solution
Put $h=x-1/\sqrt2$ and $k=y-1/(2\sqrt2)$. At the expansion point the exponent is one and the <gradient> is $(\sqrt2e,\sqrt2e)$. The first-order <multivariable Taylor expansion> is
$$
\boxed{f=e+\sqrt2e(h+k)+O(h^2+k^2).}
$$
The <directional error of a first-order Taylor approximation> is determined by the omitted terms, not just the equal first <partial derivatives>. The <Hessian matrix> at the expansion point is $e\begin{pmatrix}4&2\\2&2\end{pmatrix}$, so
$$
f-e-\sqrt2e(h+k)=e(2h^2+2hk+k^2)+O\bigl((h^2+k^2)^{3/2}\bigr).
$$
For equal small steps along the coordinate axes, the leading error is $2eh^2$ along $x$ and $ek^2$ along $y$. \b[The error grows faster along the $x$ axis], although the first-order change in the function is equal along the two axes.
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