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