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