3 weeks before
As the function doesn't behave uniformly in its domain of definition, it's broken up into three parts (look at the first line of the solution). Then the derivative has been taken in those intervals ( second step). Now there are two critical points where f'(x) changes it's behaviour. Those points are x= 1 and -1. So we test the existence of derivative at those points. As you are already familiar with the definition of f'(x) from first principle, it involves a certain limit. So the value of left and right hand derivative are found out to check the existence of limit. As we discuss Continuity and Differentiability later, you will see the conditions better. Let us know if you face any other problems in this sum.