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

Problem 2.1201 · hard

1. \[ \frac{d}{d x} \left(\frac{\ln{\left(1 - x^{2} \right)}}{2} - \frac{\ln{\left(x^{2} + 1 \right)}}{2}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\ln{\left(1 - x^{2} \right)} - \ln{\left(x^{2} + 1 \right)}\right)}{2} \]
   constant-multiplePull out the constant factor 1/2.✓ Proved
3. \[ = \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   sumApply the difference rule.✓ Proved
4. \[ = \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{\frac{d}{d x} \left(- \ln{\left(x^{2} + 1 \right)}\right)}{2} \]
   algebraDistribute the negative sign.✓ Proved
5. \[ = \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   algebraRevert to the standard subtraction form.✓ Proved
6. \[ = \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} - \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} \]
   chainApply the chain rule to the second term.✓ Proved
7. \[ = - \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} + \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \left(1 - x^{2}\right)} \]
   chainApply the chain rule to the first term.✓ Proved
8. \[ = - \frac{x}{x^{2} + 1} - \frac{x}{1 - x^{2}} \]
   derivative algebra constant-multipleDifferentiate the inner functions. Simplify the terms. Distribute the 1/2.✓ Proved
9. \[ = - \frac{x}{x^{2} + 1} + \frac{x}{x^{2} - 1} \]
   algebraSimplify the first fraction by multiplying numerator and denominator by -1.✓ Proved
10. \[ = \frac{- x \left(x^{2} - 1\right) + x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right) \left(x^{2} + 1\right)} \]
    algebraFind a common denominator.✓ Proved
11. \[ = \frac{2 x}{x^{4} - 1} \]
    algebra simplifyExpand the numerator and denominator. Combine like terms in the numerator.✓ Proved