If f and g are n-times differentiable functions, then :
Now, we would like to find out a generalized expression for the n-th derivative of fg. In order to arrive at that formulation lets calculate a few derivatives to see whether we can find a pattern:
You must have noticed a pattern in the above expressions. The coefficients seem are the one in the binomial expansion of
Therefore we can write the expression for the n-derivative of fg as the following expression:
where (i) means to differentiate i-times.
This is also known as Leibniz Formula.
** This plays an important role when we start discussing about the Associated Legendre Differential Equation.