## 2

## Successive Differentiation, Mean Value Theorems and Expansion of Functions

##### 2.1 SUCCESSIVE DIFFERENTIATION

Let *f* be a real-valued function defined on an interval [*a*, *b*]. Then, it is said to be derivable at an interior point *c* if exists. This limit, if exists, is called the *derivative* or the *differential coefficient* of the function at *x = c* and is denoted by *f* ′(*c*).

The above limit exists, if both the following limits exist and are equal:

- called the
*left*-*hand derivative* and denoted by *f* ′(*c* – 1),
- called the
*right*-*hand derivative* and denoted ...