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

Problem 2.173 · hard

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(3 x \right)} \right)}}{3}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} \right)}}{3} \]
   constantPull out the constant factor.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \sin{\left(3 x \right)}}{3 \sin{\left(3 x \right)}} \]
   chainApply the chain rule to the logarithm.✓ Proved
4. \[ = - \frac{\cos{\left(3 x \right)} \frac{d}{d x} 3 x}{3 \sin{\left(3 x \right)}} \]
   chainApply the chain rule to the sine function.✓ Proved
5. \[ = - \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} \]
   derivative algebra algebraDifferentiate the innermost function 3*x. Cancel the 3 in the numerator and denominator. Combine the terms into a single fraction.✓ Proved
6. \[ = - \cot{\left(3 x \right)} \]
   simplifyUse the trigonometric identity cot(u) = cos(u)/sin(u).✓ Proved