Derivada de $$$\ln\left(1 - x^{2}\right)$$$

La función $$$\ln\left(1 - x^{2}\right)$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \ln\left(u\right)$$$ y $$$g{\left(x \right)} = 1 - x^{2}$$$.

Aplica la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(1 - x^{2}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(1 - x^{2}\right)\right)}$$

La derivada del logaritmo natural es $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(1 - x^{2}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(1 - x^{2}\right)$$

Volver a la variable original:

$$\frac{\frac{d}{dx} \left(1 - x^{2}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(1 - x^{2}\right)}{{\color{red}\left(1 - x^{2}\right)}}$$

La derivada de una suma/diferencia es la suma/diferencia de las derivadas:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(1 - x^{2}\right)\right)}}{1 - x^{2}} = \frac{{\color{red}\left(\frac{d}{dx} \left(1\right) - \frac{d}{dx} \left(x^{2}\right)\right)}}{1 - x^{2}}$$

Aplica la regla de la potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:

$$\frac{- {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(1\right)}{1 - x^{2}} = \frac{- {\color{red}\left(2 x\right)} + \frac{d}{dx} \left(1\right)}{1 - x^{2}}$$

La derivada de una constante es $$$0$$$:

$$\frac{- 2 x + {\color{red}\left(\frac{d}{dx} \left(1\right)\right)}}{1 - x^{2}} = \frac{- 2 x + {\color{red}\left(0\right)}}{1 - x^{2}}$$

Simplificar:

$$- \frac{2 x}{1 - x^{2}} = \frac{2 x}{x^{2} - 1}$$

Por lo tanto, $$$\frac{d}{dx} \left(\ln\left(1 - x^{2}\right)\right) = \frac{2 x}{x^{2} - 1}$$$.