Derivative of \( \displaystyle \ln{\left(\frac{3 x}{\sqrt{9 x^{2} + 1}} \right)} \)

Problem 2.1424 · hard

1. \[ \frac{d}{d x} \ln{\left(\frac{3 x}{\sqrt{9 x^{2} + 1}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(\ln{\left(3 x \right)} - \ln{\left(\sqrt{9 x^{2} + 1} \right)}\right) \]
   simplify simplifyUse logarithm properties to split the quotient. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{d}{d x} \left(\ln{\left(3 x \right)} - \frac{\ln{\left(9 x^{2} + 1 \right)}}{2}\right) \]
   algebraMove the exponent to the front.✓ Proved
4. \[ = \frac{d}{d x} \ln{\left(3 x \right)} - \frac{d}{d x} \frac{\ln{\left(9 x^{2} + 1 \right)}}{2} \]
   sumApply the sum rule for derivatives.✓ Proved
5. \[ = \frac{d}{d x} \ln{\left(3 x \right)} - \frac{\frac{d}{d x} \ln{\left(9 x^{2} + 1 \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
6. \[ = \frac{d}{d x} \ln{\left(3 x \right)} - \frac{\frac{d}{d x} \left(9 x^{2} + 1\right)}{2 \left(9 x^{2} + 1\right)} \]
   logarithmicApply the derivative rule for the natural logarithm.✓ Proved
7. \[ = - \frac{9 x}{9 x^{2} + 1} + \frac{d}{d x} \ln{\left(3 x \right)} \]
   derivativeDifferentiate the inner polynomial.✓ Proved
8. \[ = - \frac{9 x}{9 x^{2} + 1} + \frac{1}{x} \]
   logarithmic algebraDifferentiate the first term. Simplify the second term.✓ Proved
9. \[ = \frac{1}{x \left(9 x^{2} + 1\right)} \]
   algebra algebra simplifyFind a common denominator. Simplify the numerator. Final simplification.✓ Proved