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Afgeleide van $$$\sin{\left(\ln\left(x\right) \right)}$$$
Gerelateerde rekenmachines: Rekenmachine voor logaritmisch differentiëren, Rekenmachine voor impliciete differentiatie met stappen
Uw invoer
Bepaal $$$\frac{d}{dx} \left(\sin{\left(\ln\left(x\right) \right)}\right)$$$.
Oplossing
De functie $$$\sin{\left(\ln\left(x\right) \right)}$$$ is de samenstelling $$$f{\left(g{\left(x \right)} \right)}$$$ van twee functies $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ en $$$g{\left(x \right)} = \ln\left(x\right)$$$.
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:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(\ln\left(x\right) \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$De afgeleide van de sinus is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(\ln\left(x\right)\right)$$Keer terug naar de oorspronkelijke variabele:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\ln\left(x\right)\right) = \cos{\left({\color{red}\left(\ln\left(x\right)\right)} \right)} \frac{d}{dx} \left(\ln\left(x\right)\right)$$De afgeleide van de natuurlijke logaritme is $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$\cos{\left(\ln\left(x\right) \right)} {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = \cos{\left(\ln\left(x\right) \right)} {\color{red}\left(\frac{1}{x}\right)}$$Dus, $$$\frac{d}{dx} \left(\sin{\left(\ln\left(x\right) \right)}\right) = \frac{\cos{\left(\ln\left(x\right) \right)}}{x}$$$.
Antwoord
$$$\frac{d}{dx} \left(\sin{\left(\ln\left(x\right) \right)}\right) = \frac{\cos{\left(\ln\left(x\right) \right)}}{x}$$$A