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Tweede afgeleide van $$$x^{3}$$$
Gerelateerde rekenmachines: Afgeleide rekenmachine, Rekenmachine voor logaritmisch differentiëren
Uw invoer
Bepaal $$$\frac{d^{2}}{dx^{2}} \left(x^{3}\right)$$$.
Oplossing
Bepaal de eerste afgeleide $$$\frac{d}{dx} \left(x^{3}\right)$$$
Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = 3$$$:
$${\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} = {\color{red}\left(3 x^{2}\right)}$$Dus, $$$\frac{d}{dx} \left(x^{3}\right) = 3 x^{2}$$$.
Vervolgens, $$$\frac{d^{2}}{dx^{2}} \left(x^{3}\right) = \frac{d}{dx} \left(3 x^{2}\right)$$$
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 $$$c = 3$$$ en $$$f{\left(x \right)} = x^{2}$$$:
$${\color{red}\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} = {\color{red}\left(3 \frac{d}{dx} \left(x^{2}\right)\right)}$$Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = 2$$$:
$$3 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = 3 {\color{red}\left(2 x\right)}$$Dus, $$$\frac{d}{dx} \left(3 x^{2}\right) = 6 x$$$.
Daarom geldt $$$\frac{d^{2}}{dx^{2}} \left(x^{3}\right) = 6 x$$$.
Antwoord
$$$\frac{d^{2}}{dx^{2}} \left(x^{3}\right) = 6 x$$$A