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

Problem 2.348 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{x}{\sqrt{1 - x^{2}}} \right)} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(\ln{\left(x \right)} - \ln{\left(\sqrt{1 - x^{2}} \right)}\right) \]
   logarithmic algebraUse the property log(a/b) = log(a) - log(b). Rewrite the square root as a power.✓ Proved
3. \[ = \frac{d}{d x} \left(\ln{\left(x \right)} - \frac{\ln{\left(1 - x^{2} \right)}}{2}\right) \]
   constant-multiplePull out the constant factor 1/2.✓ Proved
4. \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{d}{d x} \frac{\ln{\left(1 - x^{2} \right)}}{2} \]
   sumApply the difference rule for derivatives.✓ Proved
5. \[ = - \frac{d}{d x} \frac{\ln{\left(1 - x^{2} \right)}}{2} + \frac{1}{x} \]
   derivativeDifferentiate the first term.✓ Proved
6. \[ = - \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{1}{x} \]
   constant-multiplePull out the constant factor 1/2 from the second term.✓ Proved
7. \[ = - \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \left(1 - x^{2}\right)} + \frac{1}{x} \]
   chainApply the chain rule to the logarithm.✓ Proved
8. \[ = \frac{x}{1 - x^{2}} + \frac{1}{x} \]
   derivative algebraDifferentiate the inner function 1 - x**2. Simplify the signs and constants.✓ Proved
9. \[ = \frac{1}{x \left(1 - x^{2}\right)} \]
   algebra simplifyFind a common denominator. Simplify the numerator.✓ Proved