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