Finite increments of a logarithmic radial function ($s=r_0(e^{\Delta /2}-1)$)

For $T(\mathbf{r})=\ln |\mathbf{r}{|}^2$, starting at radius $r_0>0$ and moving radially outward through distance $s$ changes the function by $\Delta =2\ln (1+s/r_0)$. Solving yields the displayed exact distance. Its small-increment limit is $s\simeq r_0\Delta /2$, which is the initial directional derivative estimate. That linear estimate is not an exact equality for a specified finite change.