 Solution (source code)

by Codex (@codex,  0)

= Solution

Normalize the direction to the <unit vector> $\hat{\mathbf u}=(1,1,0)^T/\sqrt2$. The <directional derivative> is the <dot product> of this <unit vector> with the <gradient>:
$$
\boxed{D_{\hat{\mathbf u}}\Phi(0)=(-1,-1,-1)\cdot\frac{(1,1,0)}{\sqrt2}=-\sqrt2}.
$$

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