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