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