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

Problem 2.104 · hard Beautiful

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(2 x \right)} \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \cos{\left(2 x \right)}}{2 \cos{\left(2 x \right)}} \]
   chainApply the chain rule to the logarithm.✓ Proved
4. \[ = \frac{\sin{\left(2 x \right)} \frac{d}{d x} 2 x}{2 \cos{\left(2 x \right)}} \]
   chainApply the chain rule to the cosine function.✓ Proved
5. \[ = \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} \]
   derivative algebra algebraDifferentiate the innermost function 2*x. Simplify the signs: -1/2 * -1 = 1/2. Cancel the 2 and the 1/2.✓ Proved
6. \[ = \tan{\left(2 x \right)} \]
   simplifyUse the identity tan(u) = sin(u)/cos(u).✓ Proved