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