= Solution
At $(x,0,z)$ the <gradient> is $(2x,0,2z)$. The outward radial unit <vector> is $(x,0,z)/\sqrt{x^2+z^2}$ when $(x,z)\ne(0,0)$. Taking its scalar product with the <gradient> gives the radial <directional derivative>
$$
\boxed{\frac{\partial f}{\partial r}=2\sqrt{x^2+z^2}.}
$$
Equivalently, on this plane $f=r^2$. At the origin the radial unit vector is undefined, but every unit <directional derivative> is zero. Differentiating along the unnormalized scaling path $(tx,0,tz)$ instead gives $2(x^2+z^2)$ at $t=1$; that is a rate per scaling parameter, not per unit radial distance.
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