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