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

Problem 2.350 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{\sqrt{x^{2} - 1}}{\sqrt{1 - x^{2}}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \ln{\left(\sqrt{1 - x^{2}} \right)} + \ln{\left(\sqrt{x^{2} - 1} \right)}\right) \]
   logarithmic rewriteUse the property log(a/b) = log(a) - log(b). Rewrite square roots as powers.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{\ln{\left(1 - x^{2} \right)}}{2} + \frac{\ln{\left(x^{2} - 1 \right)}}{2}\right) \]
   constant-multiplePull out the constant factor 1/2.✓ Proved
4. \[ = - \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(x^{2} - 1 \right)}}{2} \]
   sumApply the derivative to each term separately.✓ Proved
5. \[ = \frac{\frac{d}{d x} \left(x^{2} - 1\right)}{2 \left(x^{2} - 1\right)} - \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \left(1 - x^{2}\right)} \]
   chainApply the chain rule to the logarithmic terms.✓ Proved
6. \[ = \frac{x}{x^{2} - 1} + \frac{x}{1 - x^{2}} \]
   derivative simplifyDifferentiate the inner functions. Simplify the coefficients and signs.✓ Proved
7. \[ = 0 \]
   algebra simplifyRewrite the second term to have a common denominator. The terms cancel out to zero.✓ Proved