I got the following question. Can anyone help me? Thanks! Y is a smooth function of X. and X is a function of r, which kinks at r=r0, but is smooth everywhere else. I was try to compute dY/dX at X=X(r0). If dX/dr exists at r=r0, then we can simply apply the chain rule to get dY/dX=(dY/dr)/(dX/dr) evaluated at r=r0.
I was wondering if we can simply do dY/dX = lim_e->0 (Y(r0+e)-Y(r0-e))/(X(r0+e)-X(r0-e))=(Y'(r0+)-Y'(r0-))/(X'(r0+)-X'(r0-)), where Y'(r0+) and Y'(r0-) are right and left derivative of Y w.r.t. r at r=r0, and similarly for X'(r0+) and X'(r0-))