Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(2 x - 1 \right)} \right)}}{2} \)

Problem 2.1510 · hard

1. \[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(2 x - 1 \right)} \right)}}{2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(2 x - 1 \right)} \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{\frac{d}{d x} \cos{\left(2 x - 1 \right)}}{2 \cos{\left(2 x - 1 \right)}} \]
   logarithmicApply the derivative rule for the natural logarithm.✓ Proved
4. \[ = - \frac{\sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{2 \cos{\left(2 x - 1 \right)}} \]
   chainApply the chain rule to the cosine function.✓ Proved
5. \[ = - \frac{\sin{\left(2 x - 1 \right)}}{\cos{\left(2 x - 1 \right)}} \]
   derivative algebraDifferentiate the inner linear function. Simplify the expression by multiplying the constants.✓ Proved
6. \[ = - \tan{\left(2 x - 1 \right)} \]
   simplifyUse the trigonometric identity sin(u)/cos(u) = tan(u).✓ Proved