The method
Suppose you wish to calculate the nth derivative of the product of two functions and . There is a general rule, called Leibniz’s rule, which gives a slightly easier way of computing it, without the process of calculating every step. The rule states that
.
So the nth derivative is the sum of n+1 terms, with the coefficients given by the nth line of Pascal’s triangle. This is very much like the binomial theorem, which states that, given two numbers and ,
.
An example
Suppose we wish to calculate . We will still need to calculate the first 5 derivatives of both functions, but that is pretty straight forward. We also calculate the 5th line of Pascal’s triangle.


So, our derivative is simply multiplying the nth coefficient with the nth derivative of and the nth derivate of , but counting this last one backwards, starting from the bottom of the table.
.
Why does it work?
The idea behind both “rules” is that we apply an operation to a general term, which we shall call the term . The operation is such that:
In particular, we have:
1) In the binomial theorem,
and F is multiplying by :
2) In Leibnitz’s rule,
and F is the derivative operator:
Here, stands for standard addition. We must also state the initial condition,
So, we have the following theorem.
1) If is a vector space and for every nonnegative m,n;
2) If there is an operation such that ,
then:
.
This generalizes both rules.
The idea of the proof was found here: http://math.stackexchange.com/questions/135510/howisleibnizsruleforthederivativeofaproductrelatedtothebinomialfo .