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Tweede afgeleide van $$$\sqrt{x}$$$
Gerelateerde rekenmachines: Afgeleide rekenmachine, Rekenmachine voor logaritmisch differentiëren
Uw invoer
Bepaal $$$\frac{d^{2}}{dx^{2}} \left(\sqrt{x}\right)$$$.
Oplossing
Bepaal de eerste afgeleide $$$\frac{d}{dx} \left(\sqrt{x}\right)$$$
Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = \frac{1}{2}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sqrt{x}\right)\right)} = {\color{red}\left(\frac{1}{2 \sqrt{x}}\right)}$$Dus, $$$\frac{d}{dx} \left(\sqrt{x}\right) = \frac{1}{2 \sqrt{x}}$$$.
Vervolgens, $$$\frac{d^{2}}{dx^{2}} \left(\sqrt{x}\right) = \frac{d}{dx} \left(\frac{1}{2 \sqrt{x}}\right)$$$
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 $$$c = \frac{1}{2}$$$ en $$$f{\left(x \right)} = \frac{1}{\sqrt{x}}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{2 \sqrt{x}}\right)\right)} = {\color{red}\left(\frac{\frac{d}{dx} \left(\frac{1}{\sqrt{x}}\right)}{2}\right)}$$Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = - \frac{1}{2}$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(\frac{1}{\sqrt{x}}\right)\right)}}{2} = \frac{{\color{red}\left(- \frac{1}{2 x^{\frac{3}{2}}}\right)}}{2}$$Dus, $$$\frac{d}{dx} \left(\frac{1}{2 \sqrt{x}}\right) = - \frac{1}{4 x^{\frac{3}{2}}}$$$.
Daarom geldt $$$\frac{d^{2}}{dx^{2}} \left(\sqrt{x}\right) = - \frac{1}{4 x^{\frac{3}{2}}}$$$.
Antwoord
$$$\frac{d^{2}}{dx^{2}} \left(\sqrt{x}\right) = - \frac{1}{4 x^{\frac{3}{2}}}$$$A