Afgeleide van $$$\ln^{3}\left(u\right)$$$
Gerelateerde rekenmachines: Rekenmachine voor logaritmisch differentiëren, Rekenmachine voor impliciete differentiatie met stappen
Uw invoer
Bepaal $$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right)$$$.
Oplossing
De functie $$$\ln^{3}\left(u\right)$$$ is de samenstelling $$$f{\left(g{\left(u \right)} \right)}$$$ van twee functies $$$f{\left(v \right)} = v^{3}$$$ en $$$g{\left(u \right)} = \ln\left(u\right)$$$.
Pas de kettingregel $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$ toe:
$${\color{red}\left(\frac{d}{du} \left(\ln^{3}\left(u\right)\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(v^{3}\right) \frac{d}{du} \left(\ln\left(u\right)\right)\right)}$$Pas de machtsregel $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$ toe met $$$n = 3$$$:
$${\color{red}\left(\frac{d}{dv} \left(v^{3}\right)\right)} \frac{d}{du} \left(\ln\left(u\right)\right) = {\color{red}\left(3 v^{2}\right)} \frac{d}{du} \left(\ln\left(u\right)\right)$$Keer terug naar de oorspronkelijke variabele:
$$3 {\color{red}\left(v\right)}^{2} \frac{d}{du} \left(\ln\left(u\right)\right) = 3 {\color{red}\left(\ln\left(u\right)\right)}^{2} \frac{d}{du} \left(\ln\left(u\right)\right)$$De afgeleide van de natuurlijke logaritme is $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$$3 \ln^{2}\left(u\right) {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} = 3 \ln^{2}\left(u\right) {\color{red}\left(\frac{1}{u}\right)}$$Dus, $$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right) = \frac{3 \ln^{2}\left(u\right)}{u}$$$.
Antwoord
$$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right) = \frac{3 \ln^{2}\left(u\right)}{u}$$$A