Derivative of \( \displaystyle - \frac{\ln{\left(x^{2} - 1 \right)}}{2} + \frac{\ln{\left(x^{2} + 1 \right)}}{2} \)

Problem 2.1262 · hard

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(x^{2} - 1 \right)}}{2} + \frac{\ln{\left(x^{2} + 1 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(- \ln{\left(x^{2} - 1 \right)} + \ln{\left(x^{2} + 1 \right)}\right)}{2} \]
   constant-multipleFactor out the constant 1/2.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(- \ln{\left(x^{2} - 1 \right)}\right)}{2} + \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   sumApply the sum rule to the terms inside.✓ Proved
4. \[ = - \frac{\frac{d}{d x} \ln{\left(x^{2} - 1 \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   constant-multipleFactor out the negative sign from the first term.✓ Proved
5. \[ = \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} - \frac{\frac{d}{d x} \left(x^{2} - 1\right)}{2 \left(x^{2} - 1\right)} \]
   chainApply the chain rule to both logarithmic terms.✓ Proved
6. \[ = \frac{x}{x^{2} + 1} - \frac{x}{x^{2} - 1} \]
   power algebra constant-multipleDifferentiate the inner quadratic functions. Multiply the terms by the derivatives. Distribute the 1/2 into the parentheses.✓ Proved
7. \[ = \frac{x \left(x^{2} - 1\right) - x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right) \left(x^{2} + 1\right)} \]
   algebraFind a common denominator to combine the fractions.✓ Proved
8. \[ = - \frac{2 x}{x^{4} - 1} \]
   algebra simplifyExpand the numerator and the denominator. Combine like terms in the numerator.✓ Proved