Derivative of \( \displaystyle - \frac{\ln{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)}}{2} \)

Problem 2.1519 · hard

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)}}{2}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)}\right)}{2 \left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)}\right)} \]
   logarithmicApply the chain rule for the natural logarithm.✓ Proved
4. \[ = - \frac{\frac{d}{d x} \tan{\left(2 x \right)} + \frac{d}{d x} \sec{\left(2 x \right)}}{2 \left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)}\right)} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = - \frac{2 \tan{\left(2 x \right)} \sec{\left(2 x \right)} + 2 \sec^{2}{\left(2 x \right)}}{2 \left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)}\right)} \]
   trigApply the derivatives of tangent and secant.✓ Proved
6. \[ = - \frac{\tan{\left(2 x \right)} \sec{\left(2 x \right)} + \sec^{2}{\left(2 x \right)}}{\tan{\left(2 x \right)} + \sec{\left(2 x \right)}} \]
   constant-multipleFactor out the constant 2.✓ Proved
7. \[ = \frac{- \tan{\left(2 x \right)} \sec{\left(2 x \right)} - \sec^{2}{\left(2 x \right)}}{\tan{\left(2 x \right)} + \sec{\left(2 x \right)}} \]
   algebraSimplify the expression by canceling the 2 and the 1/2.✓ Proved
8. \[ = - \sec{\left(2 x \right)} \]
   algebra simplifyFactor out sec(2*x) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved