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Andra derivatan av $$$\frac{1}{x^{2}}$$$
Relaterade kalkylatorer: Derivata-beräknare, Kalkylator för logaritmisk derivering
Din inmatning
Bestäm $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x^{2}}\right)$$$.
Lösning
Bestäm den första derivatan $$$\frac{d}{dx} \left(\frac{1}{x^{2}}\right)$$$
Tillämpa potensregeln $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ med $$$n = -2$$$:
$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)} = {\color{red}\left(- \frac{2}{x^{3}}\right)}$$Alltså, $$$\frac{d}{dx} \left(\frac{1}{x^{2}}\right) = - \frac{2}{x^{3}}$$$.
Därefter, $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x^{2}}\right) = \frac{d}{dx} \left(- \frac{2}{x^{3}}\right)$$$
Tillämpa konstantfaktorregeln $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ med $$$c = -2$$$ och $$$f{\left(x \right)} = \frac{1}{x^{3}}$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \frac{2}{x^{3}}\right)\right)} = {\color{red}\left(- 2 \frac{d}{dx} \left(\frac{1}{x^{3}}\right)\right)}$$Tillämpa potensregeln $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ med $$$n = -3$$$:
$$- 2 {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{3}}\right)\right)} = - 2 {\color{red}\left(- \frac{3}{x^{4}}\right)}$$Alltså, $$$\frac{d}{dx} \left(- \frac{2}{x^{3}}\right) = \frac{6}{x^{4}}$$$.
Således, $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x^{2}}\right) = \frac{6}{x^{4}}$$$.
Svar
$$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x^{2}}\right) = \frac{6}{x^{4}}$$$A