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