Define and prove the formula for the th derivative of ,
Proof. The proof is by induction. For the case we need to take two derivatives of (since ).
So, the formula holds for the case . Assume then that the formula is true for some positive integer . Then the inductive hypothesis is that,
Now, we want to take two derivatives (to get a formula for ).
Now, we take another derivative,
Thus, the formula holds for . Hence, it holds for all positive integers