Derivative of \( \displaystyle \frac{\ln{\left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)} \right)}}{3} \)

Problem 2.150 · hard

1. \[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)} \right)}}{3} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)} \right)}}{3} \]
   constant-multiplePull out the constant factor 1/3.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)}\right)}{3 \left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)}\right)} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = \frac{\frac{d}{d x} \tan{\left(3 x - 3 \right)} + \frac{d}{d x} \sec{\left(3 x - 3 \right)}}{3 \left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)}\right)} \]
   sumApply the sum rule to the inner expression.✓ Proved
5. \[ = \frac{3 \tan{\left(3 x - 3 \right)} \sec{\left(3 x - 3 \right)} + 3 \sec^{2}{\left(3 x - 3 \right)}}{3 \left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)}\right)} \]
   chainDifferentiate the tangent and secant terms using the chain rule.✓ Proved
6. \[ = \frac{\tan{\left(3 x - 3 \right)} \sec{\left(3 x - 3 \right)} + \sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)}} \]
   constant-multiple simplifyFactor out the constant 3 from the derivative sum. Cancel the 1/3 and 3, then simplify the expression.✓ Proved
7. \[ = \sec{\left(3 x - 3 \right)} \]
   algebra simplifyFactor out sec(3*x - 3) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved