Question

How to differentiate |x-1| ?

Answer 1

for differentiate such type of questions you need to prove LHD=RHD

Answer 2

To differentiate |x-1|

we have to check whether L.H.D = R.H.D
for any point lets take an example for x=a

take L.H.D

lim h->0- f(x+h) - f(x) / h

(i will not write "lim h->0-" it will be understood)

f(a+h) -f(a) /h

|a-1+h| - |a-1| / h. {by puttin x=a in x-1 we get a-1 so put that in f(a+h)}

|a-1| + |h| - |a-1| /h
|h|/h

-h/h
= -1. (for h is less than 0)

similarly solve this for R.H.D. you will get
the same step

|h|/h
=1. (as h is greater than 0).

so L.H.D is not equal to R.H.D so |x-1| is not differentiable.

hope u understand.
please let me know if u dont understand any of the step.
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