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Afgeleide van $$$- 2 e^{t} \sin{\left(t \right)}$$$
Gerelateerde rekenmachines: Rekenmachine voor logaritmisch differentiëren, Rekenmachine voor impliciete differentiatie met stappen
Uw invoer
Bepaal $$$\frac{d}{dt} \left(- 2 e^{t} \sin{\left(t \right)}\right)$$$.
Oplossing
Pas de regel van de constante factor $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ toe met $$$c = -2$$$ en $$$f{\left(t \right)} = e^{t} \sin{\left(t \right)}$$$:
$${\color{red}\left(\frac{d}{dt} \left(- 2 e^{t} \sin{\left(t \right)}\right)\right)} = {\color{red}\left(- 2 \frac{d}{dt} \left(e^{t} \sin{\left(t \right)}\right)\right)}$$Pas de productregel $$$\frac{d}{dt} \left(f{\left(t \right)} g{\left(t \right)}\right) = \frac{d}{dt} \left(f{\left(t \right)}\right) g{\left(t \right)} + f{\left(t \right)} \frac{d}{dt} \left(g{\left(t \right)}\right)$$$ toe op $$$f{\left(t \right)} = e^{t}$$$ en $$$g{\left(t \right)} = \sin{\left(t \right)}$$$:
$$- 2 {\color{red}\left(\frac{d}{dt} \left(e^{t} \sin{\left(t \right)}\right)\right)} = - 2 {\color{red}\left(\frac{d}{dt} \left(e^{t}\right) \sin{\left(t \right)} + e^{t} \frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)}$$De afgeleide van de exponentiële functie is $$$\frac{d}{dt} \left(e^{t}\right) = e^{t}$$$:
$$- 2 e^{t} \frac{d}{dt} \left(\sin{\left(t \right)}\right) - 2 \sin{\left(t \right)} {\color{red}\left(\frac{d}{dt} \left(e^{t}\right)\right)} = - 2 e^{t} \frac{d}{dt} \left(\sin{\left(t \right)}\right) - 2 \sin{\left(t \right)} {\color{red}\left(e^{t}\right)}$$De afgeleide van de sinus is $$$\frac{d}{dt} \left(\sin{\left(t \right)}\right) = \cos{\left(t \right)}$$$:
$$- 2 e^{t} \sin{\left(t \right)} - 2 e^{t} {\color{red}\left(\frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)} = - 2 e^{t} \sin{\left(t \right)} - 2 e^{t} {\color{red}\left(\cos{\left(t \right)}\right)}$$Vereenvoudig:
$$- 2 e^{t} \sin{\left(t \right)} - 2 e^{t} \cos{\left(t \right)} = - 2 \sqrt{2} e^{t} \sin{\left(t + \frac{\pi}{4} \right)}$$Dus, $$$\frac{d}{dt} \left(- 2 e^{t} \sin{\left(t \right)}\right) = - 2 \sqrt{2} e^{t} \sin{\left(t + \frac{\pi}{4} \right)}$$$.
Antwoord
$$$\frac{d}{dt} \left(- 2 e^{t} \sin{\left(t \right)}\right) = - 2 \sqrt{2} e^{t} \sin{\left(t + \frac{\pi}{4} \right)}$$$A