Hello!

I've run into a case, where one should calculate the directional derivative of a function $\displaystyle f$ of several variables at a point on a sphere of radius $\displaystyle R$ in the direction of a normal vector to this sphere. It is known that the function itself can be written only in terms of the length of the argument x, i.e. $\displaystyle |x|$.

Why is it actually allowed to write $\displaystyle \frac{\partial f}{\partial n}|_{|x|=R}=\pm \frac{\partial f}{\partial |x|}|_{|x|=R}$, with $\displaystyle \pm$ for the outer/inner normal vector?