Ableitung von $$$\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$
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Ihre Eingabe
Bestimme $$$\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right)$$$.
Lösung
Die Funktion $$$\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$ ist die Komposition $$$f{\left(g{\left(x \right)} \right)}$$$ der beiden Funktionen $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ und $$$g{\left(x \right)} = \frac{2 \ln\left(x\right)}{3}$$$.
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:
$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)\right)}$$Die Ableitung des Kosinus ist $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)$$Zurück zur ursprünglichen Variable:
$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right) = - \sin{\left({\color{red}\left(\frac{2 \ln\left(x\right)}{3}\right)} \right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)$$Wende die Konstantenfaktorregel $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ mit $$$c = \frac{2}{3}$$$ und $$$f{\left(x \right)} = \ln\left(x\right)$$$ an:
$$- \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)\right)} = - \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{2 \frac{d}{dx} \left(\ln\left(x\right)\right)}{3}\right)}$$Die Ableitung des natürlichen Logarithmus ist $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$- \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)}}{3} = - \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{1}{x}\right)}}{3}$$Somit gilt $$$\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right) = - \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}}{3 x}$$$.
Antwort
$$$\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right) = - \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}}{3 x}$$$A