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Funktion $$$\cos{\left(5 \theta \right)}$$$ derivaatta
Aiheeseen liittyvät laskurit: Logaritmisen derivoinnin laskin, Vaiheittainen implisiittisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d}{d\theta} \left(\cos{\left(5 \theta \right)}\right)$$$.
Ratkaisu
Funktio $$$\cos{\left(5 \theta \right)}$$$ on kahden funktion $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ ja $$$g{\left(\theta \right)} = 5 \theta$$$ yhdistelmä $$$f{\left(g{\left(\theta \right)} \right)}$$$.
Sovella ketjusääntöä $$$\frac{d}{d\theta} \left(f{\left(g{\left(\theta \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{d\theta} \left(g{\left(\theta \right)}\right)$$$:
$${\color{red}\left(\frac{d}{d\theta} \left(\cos{\left(5 \theta \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{d\theta} \left(5 \theta\right)\right)}$$Kosinin derivaatta on $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{d\theta} \left(5 \theta\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{d\theta} \left(5 \theta\right)$$Palaa alkuperäiseen muuttujaan:
$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right) = - \sin{\left({\color{red}\left(5 \theta\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right)$$Sovella vakion kerroinsääntöä $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$ käyttäen $$$c = 5$$$ ja $$$f{\left(\theta \right)} = \theta$$$:
$$- \sin{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(5 \theta\right)\right)} = - \sin{\left(5 \theta \right)} {\color{red}\left(5 \frac{d}{d\theta} \left(\theta\right)\right)}$$Sovella potenssisääntöä $$$\frac{d}{d\theta} \left(\theta^{n}\right) = n \theta^{n - 1}$$$ käyttäen $$$n = 1$$$, toisin sanoen, $$$\frac{d}{d\theta} \left(\theta\right) = 1$$$:
$$- 5 \sin{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(\theta\right)\right)} = - 5 \sin{\left(5 \theta \right)} {\color{red}\left(1\right)}$$Näin ollen, $$$\frac{d}{d\theta} \left(\cos{\left(5 \theta \right)}\right) = - 5 \sin{\left(5 \theta \right)}$$$.
Vastaus
$$$\frac{d}{d\theta} \left(\cos{\left(5 \theta \right)}\right) = - 5 \sin{\left(5 \theta \right)}$$$A