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Differentaiation

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This is not a monotonic smooth function. If f(x)=I6xI, then you have to distinguish two domains: x< 0 when
  f(x) =- 6x, and x > 0 when f(x) = 6x. 
 
                  Thus,   f'(x) = -6 for x<0    and f'(x) = 6 for x > 0
 
                  The derivative f'(x) is not a continuous function because there is no derivative at x=0 
                     
                 (the graph of f(x) =I6xI  is broken at x = 0).

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LOL:  We need to constantly remind our students to write clearly.  I don't think he meant abs(6x), but I'm not sure what "6.x" means - probably just multiplication ---->  6x

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As per definition

differentiation of f(x)= f'(x) = Limit h>0 (f(x+h) - f(x))/h ................ Limit h tends to zero

Therefore

f'(6x) = Limit h>0 (6(x+h) - 6x)/h

=> (6x + 6h - 6x)/h.................. limit h tends to zero

=> 6h/h = 6

:)

 

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