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