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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.

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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

Index

4. 1 Derivability At A Point
4. 2 Derivability In An Interval
4. 3 Derivability And Continuity Of A Function At A Point
4. 4 Some Counter Examples
4. 5 Interpretation Of Derivatives
4. 6 Theorems On Derivatives (differentiation Rules)
4. 7 Derivatives Of Standard Functions
4. 8 Derivative Of A Composite Function
4. 9 Differentiation Of Implicit Functions
4.10 Derivative Of An Inverse Function
4.11 Derivatives Of Inverse Trigonometric Functions
4.12 Derivatives Of Exponential & Logarithmic Functions
4.13 Logarithmic Differentiation
4.14 Derivatives Of Functions In Parametric Form
4.15 Higher order Derivatives

Chapter 5