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Ableitung von $$$\cos^{2}{\left(x \right)}$$$
Ähnliche Rechner: Rechner für logarithmische Differentiation, Rechner zur impliziten Differentiation mit Schritten
Ihre Eingabe
Bestimme $$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)$$$.
Lösung
Die Funktion $$$\cos^{2}{\left(x \right)}$$$ ist die Komposition $$$f{\left(g{\left(x \right)} \right)}$$$ der beiden Funktionen $$$f{\left(u \right)} = u^{2}$$$ und $$$g{\left(x \right)} = \cos{\left(x \right)}$$$.
Wende die Kettenregel $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ an:
$${\color{red}\left(\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$Wende die Potenzregel $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ mit $$$n = 2$$$ an:
$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$Zurück zur ursprünglichen Variable:
$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = 2 {\color{red}\left(\cos{\left(x \right)}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$Die Ableitung des Kosinus ist $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$2 \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = 2 \cos{\left(x \right)} {\color{red}\left(- \sin{\left(x \right)}\right)}$$Vereinfachen:
$$- 2 \sin{\left(x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)}$$Somit gilt $$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right) = - \sin{\left(2 x \right)}$$$.
Antwort
$$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right) = - \sin{\left(2 x \right)}$$$A