Derivative of xsinx by using first principle

Let . Then the derivative is given by the limit



Expand the numerator:



This can be rewritten as



So you're left with the limit



which you can split up as a sum of factored limits



The first two limits use two special limits,



while for the last, you can cancel out the factors of in the numerator and denominator. You're left with



You have and , so the derivative is