Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 6A / b / ii / 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 derivativeEquivalently, 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.