REMARK: (1) Progressive and Regressive derivatives
The progressive derivative of ‘f ’ at x = a is
given by
, h > 0 i.e. h ® 0^{+}
or x ® a^{+}
and it is denoted by R f ’ ( a ) or by f ’(a + 0) or by f ’ ( a^{+})
. It is also known as the Right hand derivative of ‘f ’ at x = a.
The regressive derivative of ’ f ’ at x = a, is given by
, h > 0 i.e. h ®
0^{}or x ® a^{}
and it is denoted by Lf ’ (a) or by f ’ ( a  0) or f " (a^{}
). It is also know as the Left hand derivative of ‘ f ’ at x = a
It is easy to see that f ’ (a) exists if and only if Rf ’ (a) and Lf ’ (a) exist and are equal.
Example 1
Consider the derivability of the function f(x) =  x  at the origin.
Left hand derivative =
=
=
= 1
Right hand derivative =
=
=
= +1
Thus f ’ (0 ^{} ) ¹ f ’ (0
^{+} ). Hence the function is not derivable at x = 0
