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Funktion $$$x^{4} \cos{\left(x \right)}$$$ derivaatta
Aiheeseen liittyvä laskin: Derivointilaskin
Syötteesi
Määritä $$$\frac{d}{dx} \left(x^{4} \cos{\left(x \right)}\right)$$$.
Ratkaisu
Olkoon $$$H{\left(x \right)} = x^{4} \cos{\left(x \right)}$$$.
Ota logaritmi yhtälön molemmilta puolilta: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(x^{4} \cos{\left(x \right)}\right)$$$.
Kirjoita oikea puoli uudelleen logaritmien ominaisuuksia käyttäen: $$$\ln\left(H{\left(x \right)}\right) = 4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)$$$.
Derivoi erikseen yhtälön molemmat puolet: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right)$$$.
Derivoi yhtälön vasen puoli.
Funktio $$$\ln\left(H{\left(x \right)}\right)$$$ on kahden funktion $$$f{\left(u \right)} = \ln\left(u\right)$$$ ja $$$g{\left(x \right)} = H{\left(x \right)}$$$ yhdistelmä $$$f{\left(g{\left(x \right)} \right)}$$$.
Sovella ketjusääntöä $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$Luonnollisen logaritmin derivaatta on $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$Palaa alkuperäiseen muuttujaan:
$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$Näin ollen, $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.
Derivoi yhtälön oikea puoli.
Summan/erotuksen derivaatta on derivaattojen summa/erotus:
$${\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\cos{\left(x \right)}\right)\right)\right)}$$Funktio $$$\ln\left(\cos{\left(x \right)}\right)$$$ on kahden funktion $$$f{\left(u \right)} = \ln\left(u\right)$$$ ja $$$g{\left(x \right)} = \cos{\left(x \right)}$$$ yhdistelmä $$$f{\left(g{\left(x \right)} \right)}$$$.
Sovella ketjusääntöä $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\cos{\left(x \right)}\right)\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x\right)\right) = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x\right)\right)$$Luonnollisen logaritmin derivaatta on $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \frac{d}{dx} \left(4 \ln\left(x\right)\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \frac{d}{dx} \left(4 \ln\left(x\right)\right)$$Palaa alkuperäiseen muuttujaan:
$$\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{\frac{d}{dx} \left(\cos{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{\frac{d}{dx} \left(\cos{\left(x \right)}\right)}{{\color{red}\left(\cos{\left(x \right)}\right)}}$$Kosinin derivaatta on $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{{\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}}{\cos{\left(x \right)}} = \frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{{\color{red}\left(- \sin{\left(x \right)}\right)}}{\cos{\left(x \right)}}$$Sovella vakion kerroinsääntöä $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ käyttäen $$$c = 4$$$ ja $$$f{\left(x \right)} = \ln\left(x\right)$$$:
$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + {\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right)\right)\right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + {\color{red}\left(4 \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$Luonnollisen logaritmin derivaatta on $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 4 {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 4 {\color{red}\left(\frac{1}{x}\right)}$$Sievennä:
$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{4}{x} = - \tan{\left(x \right)} + \frac{4}{x}$$Näin ollen, $$$\frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right) = - \tan{\left(x \right)} + \frac{4}{x}$$$.
Tästä seuraa, että $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = - \tan{\left(x \right)} + \frac{4}{x}$$$.
Siispä $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(- \tan{\left(x \right)} + \frac{4}{x}\right) H{\left(x \right)} = x^{3} \left(- x \tan{\left(x \right)} + 4\right) \cos{\left(x \right)}$$$.
Vastaus
$$$\frac{d}{dx} \left(x^{4} \cos{\left(x \right)}\right) = x^{3} \left(- x \tan{\left(x \right)} + 4\right) \cos{\left(x \right)}$$$A