# Math Help - Leibniz rule for differentiating the product, help please!

1. ## Leibniz rule for differentiating the product, help please!

If n is a nonnegative integer and functions f and g have nth-order derivatives, show that:
$(fg)^{(n)}=f^{(n)}g+\left(\begin{array}{ll}n\\1\en d{array}\right)f^{(n-1)}g'+\left(\begin{array}{ll}n\\2\end{array}\right )f^{(n-2)}g''+...+\left(\begin{array}{ll}n\\n\end{array}\ right)fg^{(n)}$
$=\sum_{k=0}^n \left(\begin{array}{ll}n\\k\end{array}\right)f^{(n-k)}g^{(k)}$
Im not well versed in binomial expansion so this is confusing to me

2. Originally Posted by binkypoo
If n is a nonnegative integer and functions f and g have nth-order derivatives, show that:
$(fg)^{(n)}=f^{(n)}g+\left(\begin{array}{ll}n\\1\en d{array}\right)f^{(n-1)}g'+\left(\begin{array}{ll}n\\2\end{array}\right )f^{(n-2)}g''+...+\left(\begin{array}{ll}n\\n\end{array}\ right)fg^{(n)}$
$=\sum_{k=0}^n \left(\begin{array}{ll}n\\k\end{array}\right)f^{(n-k)}g^{(k)}$
Im not well versed in binomial expansion so this is confusing to me

This is a simple proof by induction: do for n= 1,2 so that you'll get some insight in what's going on here.

Tonio