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