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