Differentiation:

Let f (x) be the real valued function defined on the open interval (a,b) where c ∈ (a,b) Then f (x) is said to be differentiable or derivable at the point  x = c,

 exists finitely.

This limit is called as derivative or differential coefficient of function

f(x) at x = c, and can be denoted by f'(c) or D f (c) or d/dx (f (x))_{x = c}

<=>Therefore, f (x) is differentiable at x = c

 exists finitely

Here  is called as left hand derivative of f (x) at x = c  and is denoted by f'(c^-) or LF'(c).

While ,  iis called as right hand derivative

of f (x) at x=c and can be denoted by f' (c⁺)or Rf' (c).

Therefore f (x) is differentiable at x = c.

                        Lf'(c) = Rf' (c)

If Lf' (c) ≠ Rf'(c) then f (x) is not differentiable at x = c.

Example : The  set of triplets (a, b, c)  of   real  numbers  having  a ≠0, for which the   function

                        f(x) = ,  is   differentiable, is

(A) { ( a,  1- 2a, a) / a ∈R;  a ≠ 0 }       

(B) { ( a,  1- 2a, c) /a, c ∈R;  a ≠ 0 }    

(C) { (a, b, c)/ a, b, c ∈R;  a + b+ c = 1 }

(D) { ( a,  1- 2a, 0) / a ∈R;  a ≠ 0 }

Solution:       (A) Given that the f is differentiable for all the real x 

                        =>  f  is  continuous   for all the  real  x.

                        so,  f(x) = f(1) => a + b + c  =1        .  . . . (1)

                        Also f'(x) = 

                        f' (1+) = f'(1-) => 1 =  2a + b   => b =  - 2a + 1                              . . . .   (2)

                        as  a, b, c ∈ R  and  a ≠ 0, using  (1) and  (2) 

                        =>  c = a

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