Derivative of \( \displaystyle \ln{\left(\frac{\sqrt{9 x^{2} - 1}}{\sqrt{1 - 9 x^{2}}} \right)} \)

Problem 2.349 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{\sqrt{9 x^{2} - 1}}{\sqrt{1 - 9 x^{2}}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \ln{\left(\sqrt{1 - 9 x^{2}} \right)} + \ln{\left(\sqrt{9 x^{2} - 1} \right)}\right) \]
   simplify rewriteUse the property log(a/b) = log(a) - log(b). Rewrite the square roots as exponents.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{\ln{\left(1 - 9 x^{2} \right)}}{2} + \frac{\ln{\left(9 x^{2} - 1 \right)}}{2}\right) \]
   constant-multiplePull out the constant factor 1/2.✓ Proved
4. \[ = - \frac{\frac{d}{d x} \ln{\left(1 - 9 x^{2} \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(9 x^{2} - 1 \right)}}{2} \]
   sumApply the derivative to each term separately.✓ Proved
5. \[ = \frac{\frac{d}{d x} \left(9 x^{2} - 1\right)}{2 \left(9 x^{2} - 1\right)} - \frac{\frac{d}{d x} \left(1 - 9 x^{2}\right)}{2 \left(1 - 9 x^{2}\right)} \]
   logarithmicApply the chain rule for the logarithm.✓ Proved
6. \[ = \frac{9 x}{9 x^{2} - 1} + \frac{9 x}{1 - 9 x^{2}} \]
   derivative algebraDifferentiate the inner polynomial terms. Simplify the products of the fractions.✓ Proved
7. \[ = 0 \]
   algebra simplifyRewrite the second term to have a common denominator. The terms cancel out to zero.✓ Proved