Derivado de $$$x^{6 x}$$$
Calculadora relacionada: Calculadora de derivados
Tu aportación
Encuentra $$$\frac{d}{dx} \left(x^{6 x}\right)$$$.
Solución
Sea $$$H{\left(x \right)} = x^{6 x}$$$.
Toma el logaritmo de ambos lados: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(x^{6 x}\right)$$$.
Vuelve a escribir la RHS usando las propiedades de los logaritmos: $$$\ln\left(H{\left(x \right)}\right) = 6 x \ln\left(x\right)$$$.
Derive por separado ambos lados de la ecuación: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(6 x \ln\left(x\right)\right)$$$.
Diferenciar el LHS de la ecuación.
La función $$$\ln\left(H{\left(x \right)}\right)$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \ln\left(u\right)$$$ y $$$g{\left(x \right)} = H{\left(x \right)}$$$.
Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$La derivada del logaritmo natural es $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$Vuelva a la variable anterior:
$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$Por lo tanto, $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.
Derive la RHS de la ecuación.
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 $$$c = 6$$$ y $$$f{\left(x \right)} = x \ln\left(x\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(6 x \ln\left(x\right)\right)\right)} = {\color{red}\left(6 \frac{d}{dx} \left(x \ln\left(x\right)\right)\right)}$$Aplique la regla del producto $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ con $$$f{\left(x \right)} = x$$$ y $$$g{\left(x \right)} = \ln\left(x\right)$$$:
$$6 {\color{red}\left(\frac{d}{dx} \left(x \ln\left(x\right)\right)\right)} = 6 {\color{red}\left(\frac{d}{dx} \left(x\right) \ln\left(x\right) + x \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$6 x \frac{d}{dx} \left(\ln\left(x\right)\right) + 6 \ln\left(x\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 6 x \frac{d}{dx} \left(\ln\left(x\right)\right) + 6 \ln\left(x\right) {\color{red}\left(1\right)}$$La derivada del logaritmo natural es $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$6 x {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + 6 \ln\left(x\right) = 6 x {\color{red}\left(\frac{1}{x}\right)} + 6 \ln\left(x\right)$$Por lo tanto, $$$\frac{d}{dx} \left(6 x \ln\left(x\right)\right) = 6 \ln\left(x\right) + 6$$$.
Por lo tanto, $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = 6 \ln\left(x\right) + 6$$$.
Por lo tanto, $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(6 \ln\left(x\right) + 6\right) H{\left(x \right)} = 6 x^{6 x} \left(\ln\left(x\right) + 1\right)$$$.
Respuesta
$$$\frac{d}{dx} \left(x^{6 x}\right) = 6 x^{6 x} \left(\ln\left(x\right) + 1\right)$$$A