Derivada de $$$\ln\left(6 x^{4}\right)$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dx} \left(\ln\left(6 x^{4}\right)\right)$$$.
Solución
La función $$$\ln\left(6 x^{4}\right)$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \ln\left(u\right)$$$ y $$$g{\left(x \right)} = 6 x^{4}$$$.
Aplica la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(6 x^{4}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(6 x^{4}\right)\right)}$$La derivada del logaritmo natural es $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(6 x^{4}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(6 x^{4}\right)$$Volver a la variable original:
$$\frac{\frac{d}{dx} \left(6 x^{4}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(6 x^{4}\right)}{{\color{red}\left(6 x^{4}\right)}}$$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 $$$c = 6$$$ y $$$f{\left(x \right)} = x^{4}$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(6 x^{4}\right)\right)}}{6 x^{4}} = \frac{{\color{red}\left(6 \frac{d}{dx} \left(x^{4}\right)\right)}}{6 x^{4}}$$Aplica la regla de la potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 4$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{4}\right)\right)}}{x^{4}} = \frac{{\color{red}\left(4 x^{3}\right)}}{x^{4}}$$Por lo tanto, $$$\frac{d}{dx} \left(\ln\left(6 x^{4}\right)\right) = \frac{4}{x}$$$.
Respuesta
$$$\frac{d}{dx} \left(\ln\left(6 x^{4}\right)\right) = \frac{4}{x}$$$A