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