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

Problem 2.1482 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\sqrt{\frac{x + 1}{1 - x}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \ln{\left(\left(\frac{x + 1}{1 - x}\right)^{0.5} \right)} \]
   rewriteRewrite the square root as a power.✓ Proved
3. \[ = \frac{d}{d x} 0.5 \ln{\left(\frac{x + 1}{1 - x} \right)} \]
   constant-multiplePull out the constant exponent.✓ Proved
4. \[ = 0.5 \frac{d}{d x} \left(- \ln{\left(1 - x \right)} + \ln{\left(x + 1 \right)}\right) \]
   algebraUse the property of logarithms to split the quotient.✓ Proved
5. \[ = - 0.5 \frac{d}{d x} \ln{\left(1 - x \right)} + 0.5 \frac{d}{d x} \ln{\left(x + 1 \right)} \]
   sumDistribute the derivative over the subtraction.✓ Proved
6. \[ = - 0.5 \frac{d}{d x} \ln{\left(1 - x \right)} + \frac{0.5}{x + 1} \]
   derivativeDifferentiate the first term.✓ Proved
7. \[ = \frac{0.5}{x + 1} + \frac{0.5}{1 - x} \]
   derivative algebraDifferentiate the second term using the chain rule. Simplify the signs.✓ Proved
8. \[ = \frac{1.0}{\left(1 - x\right) \left(x + 1\right)} \]
   algebraFind a common denominator.✓ Proved
9. \[ = \frac{1.0}{1 - x^{2}} \]
   simplifySimplify the numerator and denominator.✓ Proved
10. \[ = \frac{1}{1 - x^{2}} \]
    simplifyFinal simplification.✓ Proved