Afgeleide van $$$\sin{\left(\frac{\pi}{x} \right)}$$$
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Uw invoer
Bepaal $$$\frac{d}{dx} \left(\sin{\left(\frac{\pi}{x} \right)}\right)$$$.
Oplossing
De functie $$$\sin{\left(\frac{\pi}{x} \right)}$$$ is de samenstelling $$$f{\left(g{\left(x \right)} \right)}$$$ van twee functies $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ en $$$g{\left(x \right)} = \frac{\pi}{x}$$$.
Pas de kettingregel $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ toe:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(\frac{\pi}{x} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(\frac{\pi}{x}\right)\right)}$$De afgeleide van de sinus is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{\pi}{x}\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{\pi}{x}\right)$$Keer terug naar de oorspronkelijke variabele:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{\pi}{x}\right) = \cos{\left({\color{red}\left(\frac{\pi}{x}\right)} \right)} \frac{d}{dx} \left(\frac{\pi}{x}\right)$$Pas de regel van de constante factor $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ toe met $$$c = \pi$$$ en $$$f{\left(x \right)} = \frac{1}{x}$$$:
$$\cos{\left(\frac{\pi}{x} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{\pi}{x}\right)\right)} = \cos{\left(\frac{\pi}{x} \right)} {\color{red}\left(\pi \frac{d}{dx} \left(\frac{1}{x}\right)\right)}$$Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = -1$$$:
$$\pi \cos{\left(\frac{\pi}{x} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)} = \pi \cos{\left(\frac{\pi}{x} \right)} {\color{red}\left(- \frac{1}{x^{2}}\right)}$$Dus, $$$\frac{d}{dx} \left(\sin{\left(\frac{\pi}{x} \right)}\right) = - \frac{\pi \cos{\left(\frac{\pi}{x} \right)}}{x^{2}}$$$.
Antwoord
$$$\frac{d}{dx} \left(\sin{\left(\frac{\pi}{x} \right)}\right) = - \frac{\pi \cos{\left(\frac{\pi}{x} \right)}}{x^{2}}$$$A