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Ableitung von $$$x^{4} \cos{\left(x \right)}$$$
Verwandter Rechner: Ableitungsrechner
Ihre Eingabe
Bestimme $$$\frac{d}{dx} \left(x^{4} \cos{\left(x \right)}\right)$$$.
Lösung
Sei $$$H{\left(x \right)} = x^{4} \cos{\left(x \right)}$$$.
Logarithmieren Sie beide Seiten: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(x^{4} \cos{\left(x \right)}\right)$$$.
Schreibe die rechte Seite mithilfe der Logarithmengesetze um: $$$\ln\left(H{\left(x \right)}\right) = 4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)$$$.
Leite beide Seiten der Gleichung getrennt ab: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right)$$$.
Leite die linke Seite der Gleichung ab.
Die Funktion $$$\ln\left(H{\left(x \right)}\right)$$$ ist die Komposition $$$f{\left(g{\left(x \right)} \right)}$$$ der beiden Funktionen $$$f{\left(u \right)} = \ln\left(u\right)$$$ und $$$g{\left(x \right)} = H{\left(x \right)}$$$.
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:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$Die Ableitung des natürlichen Logarithmus ist $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$Zurück zur ursprünglichen Variable:
$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$Somit gilt $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.
Leite die rechte Seite der Gleichung ab.
Die Ableitung einer Summe/Differenz ist die Summe/Differenz der Ableitungen:
$${\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\cos{\left(x \right)}\right)\right)\right)}$$Die Funktion $$$\ln\left(\cos{\left(x \right)}\right)$$$ ist die Komposition $$$f{\left(g{\left(x \right)} \right)}$$$ der beiden Funktionen $$$f{\left(u \right)} = \ln\left(u\right)$$$ 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:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\cos{\left(x \right)}\right)\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x\right)\right) = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x\right)\right)$$Die Ableitung des natürlichen Logarithmus ist $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \frac{d}{dx} \left(4 \ln\left(x\right)\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \frac{d}{dx} \left(4 \ln\left(x\right)\right)$$Zurück zur ursprünglichen Variable:
$$\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{\frac{d}{dx} \left(\cos{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{\frac{d}{dx} \left(\cos{\left(x \right)}\right)}{{\color{red}\left(\cos{\left(x \right)}\right)}}$$Die Ableitung des Kosinus ist $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{{\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}}{\cos{\left(x \right)}} = \frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{{\color{red}\left(- \sin{\left(x \right)}\right)}}{\cos{\left(x \right)}}$$Wende die Konstantenfaktorregel $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ mit $$$c = 4$$$ und $$$f{\left(x \right)} = \ln\left(x\right)$$$ an:
$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + {\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right)\right)\right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + {\color{red}\left(4 \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$Die Ableitung des natürlichen Logarithmus ist $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 4 {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 4 {\color{red}\left(\frac{1}{x}\right)}$$Vereinfachen:
$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{4}{x} = - \tan{\left(x \right)} + \frac{4}{x}$$Somit gilt $$$\frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right) = - \tan{\left(x \right)} + \frac{4}{x}$$$.
Somit gilt $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = - \tan{\left(x \right)} + \frac{4}{x}$$$.
Daher $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(- \tan{\left(x \right)} + \frac{4}{x}\right) H{\left(x \right)} = x^{3} \left(- x \tan{\left(x \right)} + 4\right) \cos{\left(x \right)}$$$.
Antwort
$$$\frac{d}{dx} \left(x^{4} \cos{\left(x \right)}\right) = x^{3} \left(- x \tan{\left(x \right)} + 4\right) \cos{\left(x \right)}$$$A