Leibniz's Rule/One Variable/Examples/8th Derivative of x^2 sin x

Example of Use of Leibniz's Rule in One Variable
The $8$th derivative $x$ of $x^2 \sin x$ is given by:


 * $\dfrac {\d^8} {\d x^8} x^2 \sin x = x^2 \sin x - 16 x \cos x - 56 \sin x$

Proof
Leibniz's Rule in One Variable gives:


 * $\displaystyle \paren {\map f x \, \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \, \map {g^{\paren {n - k} } } x$

where $\paren n$ denotes the order of the derivative.

Here we take:
 * $\map f x = x^2$
 * $\map g x = \sin x$

We note that:

and for all $n > 2$:


 * $\dfrac {\d^n f} {d x^n} = 0$

Hence we need investigate only $\dfrac {\d^n g} {d x^n}$ where $n \in \set {6, 7, 8}$.

Thus:

Hence: