Derivative of 1/4ln(x²-1)/(x²+1)

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = \log{\left(x^{2} - 1 \right)}$ and $g{\left(x \right)} = 4 x^{2} + 4$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = x^{2} - 1$.

   2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 1\right)$:

      1. Differentiate $x^{2} - 1$ term by term:

         1. Apply the power rule: $x^{2}$ goes to $2 x$

         2. The derivative of the constant $-1$ is zero.

         The result is: $2 x$

      The result of the chain rule is:

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

   To find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $4 x^{2} + 4$ term by term:

      1. The derivative of the constant $4$ is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x^{2}$ goes to $2 x$

         So, the result is: $8 x$

      The result is: $8 x$

   Now plug in to the quotient rule:

   $\frac{- 8 x \log{\left(x^{2} - 1 \right)} + \frac{2 x \left(4 x^{2} + 4\right)}{x^{2} - 1}}{\left(4 x^{2} + 4\right)^{2}}$

2. Now simplify:

   $$\frac{x \left(x^{2} - \left(x^{2} - 1\right) \log{\left(x^{2} - 1 \right)} + 1\right)}{2 \left(x^{2} - 1\right) \left(x^{2} + 1\right)^{2}}$$

The answer is: