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Derivada de $$$\ln\left(\frac{a^{2}}{x^{2}}\right)$$$ con respecto a $$$x$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right)$$$.
Solución
La función $$$\ln\left(\frac{a^{2}}{x^{2}}\right)$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \ln\left(u\right)$$$ y $$$g{\left(x \right)} = \frac{a^{2}}{x^{2}}$$$.
Aplica la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)\right)}$$La derivada del logaritmo natural es $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)$$Volver a la variable original:
$$\frac{\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)}{{\color{red}\left(\frac{a^{2}}{x^{2}}\right)}}$$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 $$$c = a^{2}$$$ y $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:
$$\frac{x^{2} {\color{red}\left(\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)\right)}}{a^{2}} = \frac{x^{2} {\color{red}\left(a^{2} \frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}}{a^{2}}$$Aplica la regla de la potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = -2$$$:
$$x^{2} {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)} = x^{2} {\color{red}\left(- \frac{2}{x^{3}}\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right) = - \frac{2}{x}$$$.
Respuesta
$$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right) = - \frac{2}{x}$$$A