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Afgeleide van $$$\ln\left(x^{2}\right)$$$
Gerelateerde rekenmachines: Rekenmachine voor logaritmisch differentiëren, Rekenmachine voor impliciete differentiatie met stappen
Uw invoer
Bepaal $$$\frac{d}{dx} \left(\ln\left(x^{2}\right)\right)$$$.
Oplossing
De functie $$$\ln\left(x^{2}\right)$$$ is de samenstelling $$$f{\left(g{\left(x \right)} \right)}$$$ van twee functies $$$f{\left(u \right)} = \ln\left(u\right)$$$ en $$$g{\left(x \right)} = x^{2}$$$.
Pas de kettingregel $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ toe:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(x^{2}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$De afgeleide van de natuurlijke logaritme is $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(x^{2}\right)$$Keer terug naar de oorspronkelijke variabele:
$$\frac{\frac{d}{dx} \left(x^{2}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(x^{2}\right)}{{\color{red}\left(x^{2}\right)}}$$Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = 2$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)}}{x^{2}} = \frac{{\color{red}\left(2 x\right)}}{x^{2}}$$Dus, $$$\frac{d}{dx} \left(\ln\left(x^{2}\right)\right) = \frac{2}{x}$$$.
Antwoord
$$$\frac{d}{dx} \left(\ln\left(x^{2}\right)\right) = \frac{2}{x}$$$A