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

Problem 2.1033 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{\sqrt{1 - x^{2}}}{x} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \ln{\left(x \right)} + \ln{\left(\sqrt{1 - x^{2}} \right)}\right) \]
   algebra 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 sum rule for derivatives.✓ Proved
5. \[ = - \frac{d}{d x} \ln{\left(x \right)} + \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} \]
   constant-multipleDistribute the derivative operator.✓ Proved
6. \[ = \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \left(1 - x^{2}\right)} - \frac{1}{x} \]
   logarithmicApply the derivative rule for the natural logarithm.✓ Proved
7. \[ = - \frac{x}{1 - x^{2}} - \frac{1}{x} \]
   derivative algebraDifferentiate the inner function 1 - x**2. Simplify the expression by multiplying 1/2 and -2*x.✓ Proved
8. \[ = - \frac{1}{x \left(1 - x^{2}\right)} \]
   algebra algebra simplifyFind a common denominator. Expand the numerator. Simplify the final expression.✓ Proved