You are trying to prove by induction that . So you assume the result for n=k, and you are trying to prove it for n=k+1 by differentiating one more time. Where you are going wrong is that you are finding the (k+1)th derivative ofthe same function. But the result for n=k+1 refers to the (k+1)th derivative of the function . You can differentiate that k+1 times by regarding it as a product (as I just indicated) and using Leibniz's rule together with the inductive hypothesis.