If is a product of functions prove that the derivatives of are given by the formula

where

is the binomial coefficient. (See the first four exercises of Section I.4.10 on page 44 of Apostol, and in particular see Exercise #4, in which we prove the binomial theorem.)

* Proof. * The proof is by induction. Letting and we use the product rule for derivatives

So, the formula is true for the case . Assume then that it is true for . Then we consider the st derivative :

Here, we use linearity of the derivative to differentiate term by term over this finite sum. This property was established in Theorem 4.1 (i) and the comments following the theorem on page 164 of Apostol. Continuing where we left off we apply the product rule,

where we’ve reindexed the first sum to run from to instead of from to . Then, we pull out the term from the first sum and the term from the second,

Now, we recall the law of Pascal’s triangle (which we proved in a previous exercise) which establishes that

Therefore, we have

Hence, the formula holds for if it holds . Therefore, we have established that it holds for all positive integers .