Derivative of \( \displaystyle - \frac{2 \ln{\left(\cos{\left(3 x \right)} \right)}}{3} \)

Problem 2.464 · hard Beautiful

1. \[ \frac{d}{d x} \left(- \frac{2 \ln{\left(\cos{\left(3 x \right)} \right)}}{3}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{2 \frac{d}{d x} \ln{\left(\cos{\left(3 x \right)} \right)}}{3} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{2 \frac{d}{d x} \cos{\left(3 x \right)}}{3 \cos{\left(3 x \right)}} \]
   chainApply the chain rule to the logarithm.✓ Proved
4. \[ = \frac{2 \sin{\left(3 x \right)} \frac{d}{d x} 3 x}{3 \cos{\left(3 x \right)}} \]
   chainApply the chain rule to the cosine function.✓ Proved
5. \[ = \frac{2 \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}} \]
   derivative algebra simplifyDifferentiate the innermost function 3*x. Multiply the constants. Simplify the expression using the tangent identity.✓ Proved
6. \[ = 2 \tan{\left(3 x \right)} \]
   simplifyFinal simplified result.✓ Proved