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