Question

differnciablity at x=2 f(x)= x+2​

Answer 1

f'(x)=2

f'(x) is constant and its value is 2 at 2 also

Answer 2

Step-by-step explanation:

hi mate

I need to determine if f(x)=|x−2| is differentiable at 2.

I was thinking I could use the definition of a derivative (f(x+h)−f(x))h but am kind of at a loss.

Your idea of considering the definition of derivative is a good one.

Indeed, not that for h>0 we have |h|=h and for h<0 it holds |h|=−h. It follows that

1=limh→0+hh=limh→0+|h|h=limh→0+f(2+h)−f(2)h

and

−1=limh→0−−hh=limh→0−|h|h=limh→0−f(2+h)−f(2)h

Hence, the limit

limh→0f(2+h)−f(2)h

does not exists and thus the function is not differentiable at 2.

I hope so it will help u