Higher Order Derivative

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Cegep1

Mathematics

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163 words

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2 minutes

Derivative of a derivative (order higher than 1)

Do not simplify when finding patterns in the formula.

[!example]- Find the formula for the $n$ ^th derivative of $f (x) = \frac{1}{x}$ .
$n$ 0 1 2 3 4 $\dots$
$f^{(x)} (x)$ $\frac{1}{x}$ $- \frac{1}{x^{2}}$ $\frac{2}{x^{3}}$ $- \frac{2 \cdot 3}{x^{4}}$ $\frac{2 \cdot 3 \cdot 4}{x^{5}}$ $\dots$
By observation, $f^{(n)} (x) = \frac{(- 1)^{n} n!}{x^{n + 1}}$ .

$n$	0	1	2	3	4	$\dots$
$f^{(x)} (x)$	$\frac{1}{x}$	$- \frac{1}{x^{2}}$	$\frac{2}{x^{3}}$	$- \frac{2 \cdot 3}{x^{4}}$	$\frac{2 \cdot 3 \cdot 4}{x^{5}}$	$\dots$

[!example]- Find the formula for the $n$ ^th derivative of $f (x) = x^{2 / 3}$ .
$n$ 0 1 2 3 4 $\dots$
$f^{(x)} (x)$ $x^{2 / 3}$ $\frac{2}{3} x^{- 1 / 3}$ $- \frac{2}{3^{2}} x^{- 4 / 3}$ $\frac{2 \cdot 4}{3^{3}} x^{- 7 / 3}$ $- \frac{2 \cdot 4 \cdot 7}{3^{4}} x^{- 10 / 3}$ $\dots$
By observation, $f^{(n)} (x) = (\prod_{i = 1}^{n} \frac{5 - 3 i}{3}) x^{(2 - 3 n) / 3}$ .
Without pi notation, $f^{(n)} (x) = \frac{2}{3^{n}} x^{- (3 n - 2) / 3} (- 1)^{n - 1} (1 \cdot 4 \cdot 7 \dots \cdot (3 n - 5))$ for $n \geq 2$ .

$n$	0	1	2	3	4	$\dots$
$f^{(x)} (x)$	$x^{2 / 3}$	$\frac{2}{3} x^{- 1 / 3}$	$- \frac{2}{3^{2}} x^{- 4 / 3}$	$\frac{2 \cdot 4}{3^{3}} x^{- 7 / 3}$	$- \frac{2 \cdot 4 \cdot 7}{3^{4}} x^{- 10 / 3}$	$\dots$

Contributors

Ajitani Hifumi

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Last edited 29 days ago

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Higher Order Derivative ​

Contributors

Changelog

Higher Order Derivative