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