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

Problem 2.657 · hard

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{6}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{6} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)}{6 \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)} \]
   logarithmicApply the chain rule for the natural logarithm.✓ Proved
4. \[ = - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(3 x + 2 \right)}}{6 \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)} \]
   sumDifferentiate the sum inside the parenthesis.✓ Proved
5. \[ = - \frac{\frac{d}{d x} \tan^{2}{\left(3 x + 2 \right)}}{6 \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)} \]
   constantThe derivative of the constant 1 is 0.✓ Proved
6. \[ = - \frac{\tan{\left(3 x + 2 \right)} \frac{d}{d x} \tan{\left(3 x + 2 \right)}}{3 \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)} \]
   powerApply the power rule to the squared tangent term.✓ Proved
7. \[ = - \frac{\tan{\left(3 x + 2 \right)} \sec^{2}{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right)}{3 \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)} \]
   trigApply the derivative rule for tangent.≈ Checked numerically
8. \[ = - \frac{\tan{\left(3 x + 2 \right)} \sec^{2}{\left(3 x + 2 \right)}}{\tan^{2}{\left(3 x + 2 \right)} + 1} \]
   chain algebra simplifyDifferentiate the inner linear function 3*x + 2. Multiply the constants together. Simplify the expression by canceling the 6 and 1/6.✓ Proved
9. \[ = - \tan{\left(3 x + 2 \right)} \]
   algebra simplifySubstitute sec(u)**2 with 1 + tan(u)**2. Cancel the common term (tan(3*x + 2)**2 + 1) from the numerator and denominator.≈ Checked numerically