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

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Example of Use of Leibniz's Rule in One Variable

The $8$th derivative with respect to $x$ 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:

\(\ds \dfrac {\d f} {d x}\) \(=\) \(\ds \dfrac \d {d x} x^2\)
\(\ds \) \(=\) \(\ds 2 x\)
\(\ds \dfrac {\d^2 f} {d x^2}\) \(=\) \(\ds \dfrac {\d^2} {d x^2} 2 x\)
\(\ds \) \(=\) \(\ds 2\)

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:

\(\ds \dfrac {\d^8 g} {d x^8}\) \(=\) \(\ds \dfrac {\d^8} {d x^8} \sin x\)
\(\ds \) \(=\) \(\ds \sin x\)


\(\ds \dfrac {\d^7 g} {d x^7}\) \(=\) \(\ds \dfrac {\d^7} {d x^7} \sin x\)
\(\ds \) \(=\) \(\ds -\cos x\)


\(\ds \dfrac {\d^6 g} {d x^6}\) \(=\) \(\ds \dfrac {\d^6} {d x^6} \sin x\)
\(\ds \) \(=\) \(\ds -\sin x\)


Hence:

\(\ds \dfrac {\d^8} {\d x^8} x^2 \sin x\) \(=\) \(\ds \sum_{k \mathop = 0}^8 \binom 8 k \map {f^{\paren k} } x \, \map {g^{\paren {8 - k} } } x\)
\(\ds \) \(=\) \(\ds \binom 8 0 x^2 \sin x + \binom 8 1 2 x \paren {-\cos x} + \binom 8 2 2 \paren {-\sin x}\)
\(\ds \) \(=\) \(\ds x^2 \sin x - 8 \times 2 x \cos x - 28 \times 2 \sin x\)
\(\ds \) \(=\) \(\ds x^2 \sin x - 16 x \cos x - 56 \sin x\)

$\blacksquare$


Sources