Differentiability
2nd Part
Differentiability
Differentiability of a function:
The
derivative
ofwith respect to x (for any particular value of x) is denoted by
orand defined as,
y f x
f x
'dy dx
lim
0 xf x x f x dy
dx
x
lim0 x
f x h f x h
Types of Differentiability of a function:
Differentiability of a function:
Left hand derivative Right hand derivative
. . lim0 h
f a h f a L H D
h
. . lim
0 hf a h f a R H D
h
Existence of Derivative:
A function
is called differentiable at
If the left hand derivative and right hand derivative both are Equal at this point that is,
and
are both exist and equal.
Existence of Derivative
. . lim0 h
f a h f a L H D
h
. . lim
0 hf a h f a R H D
h
y f x
x a
Problem
A function
is defined as follows:Discuss the differentiability at
and
.
2
1 0
0 1
1 1
x when x
f x x when x
when x x
f x
0
x x 1
Solution
Solution: Given that,
Part: For
2 1 0
0 1
1 1
x when x
f x x when x
when x x
0
0
2 2
0 2
0 2
0
0
0 0
. . lim lim 0
1 0 1
lim lim 1 1
lim lim 0
h
h
h
h
h
h
f h f
L H D
h
f h f
h h
h h
h h
h h
0 x
0
0
2
0
0
0
0 0
. . lim lim 0
0 1
lim lim 1 lim 1 1
h
h
h
h
h
f h f
R H D
h f h f
h h
h h
h h
Since R.H.D does not exist. So the function is not differentiable at
2
ndPart: For Since
Does not exist. So the
function is not
differentiable at
.
0 x 1
x
0
0
0
0
1 1
. . lim
1 1
lim lim lim 1 1
h
h
h
h
f h f
L H D
h h
h h h
0
0
0
0
1 1
. . lim
1 1
lim 1 lim 1 1
1 lim 1
1 1 1 0
1
h
h
h
h
f h f
R H D
h h
h h
h h
h
. . . .
L H D R H D
1
x
Practice
A function
is defined as follows:Discuss the differentiability at
and x =
f x
21 0
1 sin 0
2
2 2 2
when x
f x x when x
x when x
