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Tweede afgeleide van $$$\frac{1}{x}$$$
Gerelateerde rekenmachines: Afgeleide rekenmachine, Rekenmachine voor logaritmisch differentiëren
Uw invoer
Bepaal $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x}\right)$$$.
Oplossing
Bepaal de eerste afgeleide $$$\frac{d}{dx} \left(\frac{1}{x}\right)$$$
Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = -1$$$:
$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)} = {\color{red}\left(- \frac{1}{x^{2}}\right)}$$Dus, $$$\frac{d}{dx} \left(\frac{1}{x}\right) = - \frac{1}{x^{2}}$$$.
Vervolgens, $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x}\right) = \frac{d}{dx} \left(- \frac{1}{x^{2}}\right)$$$
Pas de regel van de constante factor $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ toe met $$$c = -1$$$ en $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \frac{1}{x^{2}}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}$$Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = -2$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)} = - {\color{red}\left(- \frac{2}{x^{3}}\right)}$$Dus, $$$\frac{d}{dx} \left(- \frac{1}{x^{2}}\right) = \frac{2}{x^{3}}$$$.
Daarom geldt $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x}\right) = \frac{2}{x^{3}}$$$.
Antwoord
$$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x}\right) = \frac{2}{x^{3}}$$$A