Derivative of \( \displaystyle \frac{\ln{\left(\tan^{2}{\left(3 x - 1 \right)} + 1 \right)}}{6} \)

Problem 2.156 · hard Beautiful

1. \[ \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(3 x - 1 \right)} + 1 \right)}}{6} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(3 x - 1 \right)} + 1 \right)}}{6} \]
   constantPull out the constant factor.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\tan^{2}{\left(3 x - 1 \right)} + 1\right)}{6 \left(\tan^{2}{\left(3 x - 1 \right)} + 1\right)} \]
   chainApply the chain rule to the logarithm.✓ Proved
4. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(3 x - 1 \right)}}{6 \left(\tan^{2}{\left(3 x - 1 \right)} + 1\right)} \]
   sumDifferentiate the sum inside the parenthesis.✓ Proved
5. \[ = \frac{\frac{d}{d x} \tan^{2}{\left(3 x - 1 \right)}}{6 \left(\tan^{2}{\left(3 x - 1 \right)} + 1\right)} \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = \frac{\tan{\left(3 x - 1 \right)} \frac{d}{d x} \tan{\left(3 x - 1 \right)}}{3 \left(\tan^{2}{\left(3 x - 1 \right)} + 1\right)} \]
   powerApply the power rule to the tangent squared term.✓ Proved
7. \[ = \frac{\tan{\left(3 x - 1 \right)} \sec^{2}{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{3 \left(\tan^{2}{\left(3 x - 1 \right)} + 1\right)} \]
   trigApply the chain rule to the tangent function.≈ Checked numerically
8. \[ = \frac{\tan{\left(3 x - 1 \right)} \sec^{2}{\left(3 x - 1 \right)}}{\tan^{2}{\left(3 x - 1 \right)} + 1} \]
   derivative algebra simplifyDifferentiate the inner linear function. Multiply the constants together. Simplify the expression by canceling the 6s.✓ Proved
9. \[ = \tan{\left(3 x - 1 \right)} \]
   rewrite simplifyUse the identity 1 + tan(u)^2 = sec(u)^2. Cancel the common sec(3*x - 1)^2 term.≈ Checked numerically