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Derivada de $$$- 2 e^{t} \sin{\left(t \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dt} \left(- 2 e^{t} \sin{\left(t \right)}\right)$$$.
Solución
Aplica la regla del factor constante $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ con $$$c = -2$$$ y $$$f{\left(t \right)} = e^{t} \sin{\left(t \right)}$$$:
$${\color{red}\left(\frac{d}{dt} \left(- 2 e^{t} \sin{\left(t \right)}\right)\right)} = {\color{red}\left(- 2 \frac{d}{dt} \left(e^{t} \sin{\left(t \right)}\right)\right)}$$Aplica la regla del producto $$$\frac{d}{dt} \left(f{\left(t \right)} g{\left(t \right)}\right) = \frac{d}{dt} \left(f{\left(t \right)}\right) g{\left(t \right)} + f{\left(t \right)} \frac{d}{dt} \left(g{\left(t \right)}\right)$$$ con $$$f{\left(t \right)} = e^{t}$$$ y $$$g{\left(t \right)} = \sin{\left(t \right)}$$$:
$$- 2 {\color{red}\left(\frac{d}{dt} \left(e^{t} \sin{\left(t \right)}\right)\right)} = - 2 {\color{red}\left(\frac{d}{dt} \left(e^{t}\right) \sin{\left(t \right)} + e^{t} \frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)}$$La derivada de la función exponencial es $$$\frac{d}{dt} \left(e^{t}\right) = e^{t}$$$:
$$- 2 e^{t} \frac{d}{dt} \left(\sin{\left(t \right)}\right) - 2 \sin{\left(t \right)} {\color{red}\left(\frac{d}{dt} \left(e^{t}\right)\right)} = - 2 e^{t} \frac{d}{dt} \left(\sin{\left(t \right)}\right) - 2 \sin{\left(t \right)} {\color{red}\left(e^{t}\right)}$$La derivada del seno es $$$\frac{d}{dt} \left(\sin{\left(t \right)}\right) = \cos{\left(t \right)}$$$:
$$- 2 e^{t} \sin{\left(t \right)} - 2 e^{t} {\color{red}\left(\frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)} = - 2 e^{t} \sin{\left(t \right)} - 2 e^{t} {\color{red}\left(\cos{\left(t \right)}\right)}$$Simplificar:
$$- 2 e^{t} \sin{\left(t \right)} - 2 e^{t} \cos{\left(t \right)} = - 2 \sqrt{2} e^{t} \sin{\left(t + \frac{\pi}{4} \right)}$$Por lo tanto, $$$\frac{d}{dt} \left(- 2 e^{t} \sin{\left(t \right)}\right) = - 2 \sqrt{2} e^{t} \sin{\left(t + \frac{\pi}{4} \right)}$$$.
Respuesta
$$$\frac{d}{dt} \left(- 2 e^{t} \sin{\left(t \right)}\right) = - 2 \sqrt{2} e^{t} \sin{\left(t + \frac{\pi}{4} \right)}$$$A