Afgeleide van $$$\ln\left(\frac{2 x}{3}\right)$$$
Gerelateerde rekenmachines: Rekenmachine voor logaritmisch differentiëren, Rekenmachine voor impliciete differentiatie met stappen
Uw invoer
Bepaal $$$\frac{d}{dx} \left(\ln\left(\frac{2 x}{3}\right)\right)$$$.
Oplossing
De functie $$$\ln\left(\frac{2 x}{3}\right)$$$ is de samenstelling $$$f{\left(g{\left(x \right)} \right)}$$$ van twee functies $$$f{\left(u \right)} = \ln\left(u\right)$$$ en $$$g{\left(x \right)} = \frac{2 x}{3}$$$.
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:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\frac{2 x}{3}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\frac{2 x}{3}\right)\right)}$$De afgeleide van de natuurlijke logaritme is $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\frac{2 x}{3}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\frac{2 x}{3}\right)$$Keer terug naar de oorspronkelijke variabele:
$$\frac{\frac{d}{dx} \left(\frac{2 x}{3}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(\frac{2 x}{3}\right)}{{\color{red}\left(\frac{2 x}{3}\right)}}$$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 $$$c = \frac{2}{3}$$$ en $$$f{\left(x \right)} = x$$$:
$$\frac{3 {\color{red}\left(\frac{d}{dx} \left(\frac{2 x}{3}\right)\right)}}{2 x} = \frac{3 {\color{red}\left(\frac{2 \frac{d}{dx} \left(x\right)}{3}\right)}}{2 x}$$Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = 1$$$, met andere woorden, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{x} = \frac{{\color{red}\left(1\right)}}{x}$$Dus, $$$\frac{d}{dx} \left(\ln\left(\frac{2 x}{3}\right)\right) = \frac{1}{x}$$$.
Antwoord
$$$\frac{d}{dx} \left(\ln\left(\frac{2 x}{3}\right)\right) = \frac{1}{x}$$$A