Determine which of the following are differentiable at 0.
                                  (i)


                                f(x)=x|x|


(ii) 


(iii) 


SOLUTION 


(i) the differentiability of f(x)=x|x| can be obtained by analyzing the function itself since the function is a product function. First the differentiability of x at 0 exists by using the test of differentiability I.e



So also  does the differentiability of |x| at 0 exists by using the test of differentiability. Since the both function are differentiable at 0 when treated as separate functions, then they are also differentiable at 0 when treated as a product function. 

(ii) take the derivative of f(x) 
Since f'(x)=2x for x>0, exists then it can be proven analytically by taking small values of 0.
f(0.1)=0.2, f(0.01)=0.02, f(0.001)=0.003 which satisfies x>0 but f(x)=0 at x<0 is not differentiable at 0.
(iii) take the derivative of f'(x)

you can also use analytical approach to verify the differentiability of the function.