Segunda derivada de $$$e^{5 x}$$$
Calculadoras relacionadas: Calculadora de derivados, Calculadora de diferenciación logarítmica
Tu aportación
Encuentra $$$\frac{d^{2}}{dx^{2}} \left(e^{5 x}\right)$$$.
Solución
Encuentra la primera derivada $$$\frac{d}{dx} \left(e^{5 x}\right)$$$
La función $$$e^{5 x}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = e^{u}$$$ y $$$g{\left(x \right)} = 5 x$$$.
Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{5 x}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(5 x\right)\right)}$$La derivada de la exponencial es $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(5 x\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(5 x\right)$$Vuelva a la variable anterior:
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(5 x\right) = e^{{\color{red}\left(5 x\right)}} \frac{d}{dx} \left(5 x\right)$$Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 5$$$ y $$$f{\left(x \right)} = x$$$:
$$e^{5 x} {\color{red}\left(\frac{d}{dx} \left(5 x\right)\right)} = e^{5 x} {\color{red}\left(5 \frac{d}{dx} \left(x\right)\right)}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$5 e^{5 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 5 e^{5 x} {\color{red}\left(1\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(e^{5 x}\right) = 5 e^{5 x}$$$.
A continuación, $$$\frac{d^{2}}{dx^{2}} \left(e^{5 x}\right) = \frac{d}{dx} \left(5 e^{5 x}\right)$$$
Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 5$$$ y $$$f{\left(x \right)} = e^{5 x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(5 e^{5 x}\right)\right)} = {\color{red}\left(5 \frac{d}{dx} \left(e^{5 x}\right)\right)}$$La función $$$e^{5 x}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = e^{u}$$$ y $$$g{\left(x \right)} = 5 x$$$.
Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$5 {\color{red}\left(\frac{d}{dx} \left(e^{5 x}\right)\right)} = 5 {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(5 x\right)\right)}$$La derivada de la exponencial es $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$$5 {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(5 x\right) = 5 {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(5 x\right)$$Vuelva a la variable anterior:
$$5 e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(5 x\right) = 5 e^{{\color{red}\left(5 x\right)}} \frac{d}{dx} \left(5 x\right)$$Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 5$$$ y $$$f{\left(x \right)} = x$$$:
$$5 e^{5 x} {\color{red}\left(\frac{d}{dx} \left(5 x\right)\right)} = 5 e^{5 x} {\color{red}\left(5 \frac{d}{dx} \left(x\right)\right)}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$25 e^{5 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 25 e^{5 x} {\color{red}\left(1\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(5 e^{5 x}\right) = 25 e^{5 x}$$$.
Por lo tanto, $$$\frac{d^{2}}{dx^{2}} \left(e^{5 x}\right) = 25 e^{5 x}$$$.
Respuesta
$$$\frac{d^{2}}{dx^{2}} \left(e^{5 x}\right) = 25 e^{5 x}$$$A