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4.2 Derivability in an Interval

A function ‘f’ defined on [a, b] is derivable at the end point a i.e. f ’ (a) exists if   exists

Similarly it is defined at the end point b, if
if  exists.

If a function is derivable at all points of an interval except the end points a and b, it is said to be derivable in an open interval (a, b).

A function is derivable in the closed interval [a, b] if it is derivable in the open interval (a, b) and also at the end points a and b.

Example 2 Show that the function f (x) = x2 is derivable on [0, 1]

Let x0 be any point of (0, 1), then
f ’ (x 0 ) = 

   = 2x0

At the end points, we have

 f ’ (0)    =   




f ’ (1)    =   



Thus the function is derivable in [0, 1].


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

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