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Funktion $$$\cos^{2}{\left(x \right)}$$$ derivaatta
Aiheeseen liittyvät laskurit: Logaritmisen derivoinnin laskin, Vaiheittainen implisiittisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)$$$.
Ratkaisu
Funktio $$$\cos^{2}{\left(x \right)}$$$ on kahden funktion $$$f{\left(u \right)} = u^{2}$$$ ja $$$g{\left(x \right)} = \cos{\left(x \right)}$$$ yhdistelmä $$$f{\left(g{\left(x \right)} \right)}$$$.
Sovella ketjusääntöä $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$Sovella potenssisääntöä $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$, kun $$$n = 2$$$:
$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$Palaa alkuperäiseen muuttujaan:
$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = 2 {\color{red}\left(\cos{\left(x \right)}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$Kosinin derivaatta on $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$2 \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = 2 \cos{\left(x \right)} {\color{red}\left(- \sin{\left(x \right)}\right)}$$Sievennä:
$$- 2 \sin{\left(x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)}$$Näin ollen, $$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right) = - \sin{\left(2 x \right)}$$$.
Vastaus
$$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right) = - \sin{\left(2 x \right)}$$$A