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Derivada de $$$2 e^{2 x}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dx} \left(2 e^{2 x}\right)$$$.
Solución
Aplica la regla del factor constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 2$$$ y $$$f{\left(x \right)} = e^{2 x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 e^{2 x}\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(e^{2 x}\right)\right)}$$La función $$$e^{2 x}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = e^{u}$$$ y $$$g{\left(x \right)} = 2 x$$$.
Aplica la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(e^{2 x}\right)\right)} = 2 {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(2 x\right)\right)}$$La derivada de la función exponencial es $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$$2 {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(2 x\right) = 2 {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(2 x\right)$$Volver a la variable original:
$$2 e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(2 x\right) = 2 e^{{\color{red}\left(2 x\right)}} \frac{d}{dx} \left(2 x\right)$$Aplica la regla del factor constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 2$$$ y $$$f{\left(x \right)} = x$$$:
$$2 e^{2 x} {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} = 2 e^{2 x} {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}$$Aplica la regla de la potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$4 e^{2 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 4 e^{2 x} {\color{red}\left(1\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(2 e^{2 x}\right) = 4 e^{2 x}$$$.
Respuesta
$$$\frac{d}{dx} \left(2 e^{2 x}\right) = 4 e^{2 x}$$$A