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