(a)               

        x(t)    --------LTI system--------->  y(t) 

 =>    x(t+h)  ----------------->  y(t+h)        .........(1)              [ from shift invariance property]

 &    -x(t)  ------------------->   -y(t)          ...........(2)             [ from homogeneity]

 now applying additivity in above 2 equations

    x(t+h) - x(t)   -------------------->  y(t+h) - y(t)  ..........(3)

 now applying homogeneity  in equation (3)

(1/h) [ x(t+h) - x(t) ]   ----------------->  (1/h) [ y(t+h) - y(t) ]

  x'(t)   ------------------->  y'(t)

      Differentiating both sides w. r. t.  t                  

      which is the output to the input x(t)

  ........(1)

      Let

      Hence                   a(t) = x'(t)

      We can also write  (1)   as                  

                                                                                                                           [ As Convolution is Commutative ]

       =>  y(t) =  b(t)

       => Y(t) =  y'(t)

     hence output to the input x'(t) is y'(t)

       or equivalently                    

       where

Again