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

Problem 2.721 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{\sqrt{1 - x^{2}}}{\sqrt{x^{2} - 1}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \ln{\left(\sqrt{\frac{1 - x^{2}}{x^{2} - 1}} \right)} \]
   algebra rewriteCombine the square roots into one. Rewrite the square root as an exponent.✓ Proved
3. \[ = \frac{d}{d x} \frac{\ln{\left(\frac{1 - x^{2}}{x^{2} - 1} \right)}}{2} \]
   simplifySimplify the nested logarithm.✓ Proved
4. \[ = \frac{\frac{d}{d x} \ln{\left(\frac{1 - x^{2}}{x^{2} - 1} \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
5. \[ = \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} - \frac{\frac{d}{d x} \ln{\left(x^{2} - 1 \right)}}{2} \]
   logarithmicApply the logarithm quotient rule.✓ Proved
6. \[ = \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{\frac{d}{d x} \left(- \ln{\left(x^{2} - 1 \right)}\right)}{2} \]
   algebraDistribute the negative sign.✓ Proved
7. \[ = \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(\frac{1}{x^{2} - 1} \right)}}{2} \]
   algebra rewriteRewrite the negative log as a power. Use the exponential form for the power.✓ Proved
8. \[ = \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{\frac{d}{d x} \left(- \ln{\left(x^{2} - 1 \right)}\right)}{2} \]
   simplifySimplify the power expression.✓ Proved
9. \[ = - \frac{\frac{d}{d x} \ln{\left(x^{2} - 1 \right)}}{2} + \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \left(1 - x^{2}\right)} \]
   logarithmicDifferentiate the first logarithm.✓ Proved
10. \[ = - \frac{x}{x^{2} - 1} - \frac{x}{1 - x^{2}} \]
    derivative algebraDifferentiate the remaining terms. Simplify the expression.✓ Proved
11. \[ = 0 \]
    algebra simplifyNote that 1/(1-x^2) = -1/(x^2-1). The terms cancel out to zero.✓ Proved