(a) and f
(x) respectively. The restriction in both the cases is that 'the limit must exist'. IfWe have already defined the derivative of a function f(x) at a particular point 'a' and derivative of f(x) in general for the variable x as f(a) and f(x) respectively. The restriction in both the cases is that 'the limit must exist'. If
If the above limit exists, we say the function f(x) is differentiable.
In order to test the differentiability of a function at a point, the right hand derivative and left hand derivatives are introduced as follows:Right Hand Derivative
Let f be a function of x (y=f(x)). Let a be a point in the domain of f. The RHD of f at a is defined as
where h>0, provided the limit exists.
Left Hand Derivative
The LHD of f at a is defined as
where h>0, provided the limit exists.
Note:
In the above definition, substitute a + h = x, then h = x - a as
Rf '(a) can be rewritten as
Similarly, substitute a - h = x. Lf '(a) can be written as
Differentiability at a Point
A function f(x) is said to be differentiable at a point 'a' if
(i) both Rf '(a) and Lf '(a) exists and finite.(ii) Rf '(a) = Lf '(a)
Example:
Consider the function y=|x|. This function is differentiable on (
,0) and (0,
), but not differentiable at x = 0.
Since the limit exists, f(x) is differentiable at x>0. Similarly, we can show that f(x) is differentiable at x<0.
We shall find RHD and LHD of f(x) at x = 0.
= -1
Therefore y = |x| is not differentiable at x = 0.
Relation Between Continuity and Differentiability
The following theorem say 'Differentiability implies continuity".
Theorem:
If a function f is differentiable at x = a, then it is continuous at x = a.
Proof:
f is differentiable at x = a.
Note:
Every differential function is continuous, but every continuous function is not differential.
We have already shown that y = |x| is not differentiable at x = 0. It can be easily seen that y = |x| is continuous at x = 0.