No publishing on paper. This definition permits to find a primitive without integration-summation. In the point where this differentiation takes place a function is bijective. Otherwise one of two limits has not meaning. Sush derivation is possible to call as symmetric or s-derivation.
Further derivation is understood as s-derivation.
As differentiable it is continuous and bijective. Then x x 0 is equivalent to y y 0 . Under x x 0 the limit of left part exists and finites. Each of two derivatives is finite. Consequently it can t have a zero meaning and with taking all intermediate values it has a strict sign.