Cool Calculus: Extending the Product Rule to the n-th Derivative

The n-th derivative of a product looks remarkably like a binomial expansion

We all know from high school how to get the derivative of a product of two functions u(x) and v(x):

In this article, I’m going to show an expression for the n-th derivative of a product — this expression will look remarkably like a binomial expansion. Recall the binomial expansion of an n-th power looks like this:

Another way of writing this is:

where

which is a well-known terminology for binomial coefficients.

The expression for the n-th derivative of a product which I am going to prove is as follows:

where the index (k) means k-th derivative, and where the zero-th derivative is the function itself.

Proof by induction

I’m going to use an induction argument to prove this result. Those of you who frequently read my articles will know that I like to describe induction as a neverending chain of dominos lined up to knock each other over one by one. To know that all dominos will fall, you need to know that the first one will fall (the induction start) and that whenever a domino falls it will cause the…