Well for the first function.....

ie         f(x)  = tan^-1(x)    

 

from this expression it can be deduced that f '(x) does not exist for  1  

 

But in the question it is given      |x|  1       ie         -1  x  1        

so the domain of the derivativebecomes

 

 

 

Now for the second question

ie         f (x)  = ¹/₂ |x - 1|

 

it can be derived that (^d/_dx)(|x|) = ^x/_|x|                            (*)       

                                            

 

so, for our function f '(x) = ^{(x - 1)}/_{|x - 1|}  

                                                 

 

obviously f '(x)  is undefined for  x=1

 

but from question x > 1

so the domain of the derivative is also     x > 1       or  

   

(*) nudge me if u want the proof of   f '(x)  of  f (x)=|x| 

 

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