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Afgeleide van $$$\operatorname{atan}{\left(4 x \right)}$$$
Gerelateerde rekenmachines: Rekenmachine voor logaritmisch differentiëren, Rekenmachine voor impliciete differentiatie met stappen
Uw invoer
Bepaal $$$\frac{d}{dx} \left(\operatorname{atan}{\left(4 x \right)}\right)$$$.
Oplossing
De functie $$$\operatorname{atan}{\left(4 x \right)}$$$ is de samenstelling $$$f{\left(g{\left(x \right)} \right)}$$$ van twee functies $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$ en $$$g{\left(x \right)} = 4 x$$$.
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:
$${\color{red}\left(\frac{d}{dx} \left(\operatorname{atan}{\left(4 x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) \frac{d}{dx} \left(4 x\right)\right)}$$De afgeleide van de inverse tangens is $$$\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) = \frac{1}{u^{2} + 1}$$$:
$${\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right)\right)} \frac{d}{dx} \left(4 x\right) = {\color{red}\left(\frac{1}{u^{2} + 1}\right)} \frac{d}{dx} \left(4 x\right)$$Keer terug naar de oorspronkelijke variabele:
$$\frac{\frac{d}{dx} \left(4 x\right)}{{\color{red}\left(u\right)}^{2} + 1} = \frac{\frac{d}{dx} \left(4 x\right)}{{\color{red}\left(4 x\right)}^{2} + 1}$$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 $$$c = 4$$$ en $$$f{\left(x \right)} = x$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(4 x\right)\right)}}{16 x^{2} + 1} = \frac{{\color{red}\left(4 \frac{d}{dx} \left(x\right)\right)}}{16 x^{2} + 1}$$Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = 1$$$, met andere woorden, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\frac{4 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{16 x^{2} + 1} = \frac{4 {\color{red}\left(1\right)}}{16 x^{2} + 1}$$Dus, $$$\frac{d}{dx} \left(\operatorname{atan}{\left(4 x \right)}\right) = \frac{4}{16 x^{2} + 1}$$$.
Antwoord
$$$\frac{d}{dx} \left(\operatorname{atan}{\left(4 x \right)}\right) = \frac{4}{16 x^{2} + 1}$$$A