# HOW TO CHECK DIFFERENTIABILITY OF A FUNCTION AT A POINT

## About "How to Check Differentiability of a Function at a Point"

How to Check Differentiability of a Function at a Point :

Here we are going to see how to check differentiability of a function at a point

The function is differentiable from the left and right. As in the case of the existence of limits of a function at x0, it follows that

exists if and only if both

exist and f' (x0-)  =   f' (x0+)

Hence

if and only if f' (x0-)  =   f' (x0+) . If any one of the condition fails then f'(x) is not differentiable at x0.

## How to Know If a Function is Differentiable at a Point - Examples

Question 1 :

(i)  f(x) = |x| + |x - 1| at x = 0, 1

Solution :

f(x) = |x| + |x - 1|

Check if the given function is continuous at x = 0.

If x < 0, then f(x) = -x - (x - 1)

f(x)  =  -x - x + 1

=  -2x + 1

If x > 0 and x < 1, then f(x) = x - (x - 1)

f(x)  =  x - x + 1

=  1

If x > 1, then f(x) = x + (x - 1)

f(x)  =  x + x + 1

=  2x + 1

f'(0-)  =  lim x->0- [(f(x) - f(0)) / (x - 0)]

=  lim x->0- [(-2x + 1) - 1] / x

=  lim x->0- -2x / x

=  lim x->0- -2

=  -2  -----(1)

f'(0+)  =  lim x->0+ [(f(x) - f(0)) / (x - 0)]

=  lim x->0+ [1 - 1] / x

=  lim x->0- 0 / x

=  0/0  -----(2)

f'(0-)  ≠ f'(0+

f'(1-)  =  lim x->1- [(f(x) - f(1)) / (x - 1)]

=  lim x->1- [1 - 1] / (x - 1)

=  lim x->1- 0 / (x-1)

=  0 -----(1)

f'(1+)  =  lim x->1+ [(f(x) - f(1)) / (x - 1)]

=  lim x->1+ [2x + 1 - 3] / (x - 1)

=  lim x->1+ (2x - 2) / (x - 1)

=  lim x->1+ 2(x - 1) / (x - 1)

=  2 -----(2)

f'(1-)  ≠ f'(1+

Hence the given function is not differentiable at the given points.

(ii)  f(x)  =  sin |x| at x = 0

Solution :

If x < 0, then f(x) = sin (-x)

f(x)  =  - sin x

If x > 0, then f(x) =  sin x

f (x)  =  sin x

f'(0-)  =  lim x->0- [(f(x) - f(0)) / (x - 0)]

=  lim x->0- [(-sin x) - 0] / x

=  lim x->0- -(sin x/x)

=  -1  -----(1)

f'(0+)  =  lim x->0+ [(f(x) - f(0)) / (x - 0)]

=  lim x->0+ [(sin x) - 0] / x

=  lim x->0- (sin x/x)

=  1  -----(2)

f'(0-)  ≠ f'(0+

Hence the given function is not differentiable at x = 0.

After having gone through the stuff given above, we hope that the students would have understood, "How to Check Differentiability of a Function at a Point"

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