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

Problem 2.678 · hard

1. \[ \frac{d}{d x} \left(- \frac{2 \ln{\left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)} \right)}}{3}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{2 \frac{d}{d x} \ln{\left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)} \right)}}{3} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{2 \frac{d}{d x} \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)}{3 \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = - \frac{2 \left(\frac{d}{d x} \cot{\left(3 x - 3 \right)} + \frac{d}{d x} \csc{\left(3 x - 3 \right)}\right)}{3 \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)} \]
   sumApply the sum rule to the inner terms.✓ Proved
5. \[ = - \frac{2 \left(- \cot{\left(3 x - 3 \right)} \csc{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right) - \csc^{2}{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)\right)}{3 \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)} \]
   trigDifferentiate the trigonometric functions.✓ Proved
6. \[ = - \frac{2 \left(- 3 \cot{\left(3 x - 3 \right)} \csc{\left(3 x - 3 \right)} - 3 \csc^{2}{\left(3 x - 3 \right)}\right)}{3 \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)} \]
   chain algebraDifferentiate the inner linear function 3*x - 3. Factor out the common term 3.✓ Proved
7. \[ = \frac{2 \cot{\left(3 x - 3 \right)} \csc{\left(3 x - 3 \right)} + 2 \csc^{2}{\left(3 x - 3 \right)}}{\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}} \]
   algebraSimplify the expression by canceling terms.✓ Proved
8. \[ = 2 \csc{\left(3 x - 3 \right)} \]
   algebra simplifyFactor out csc(3*x - 3) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved