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

Problem 2.1828 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
   chainApply the chain rule for the natural logarithm.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(- \cot{\left(3 x \right)}\right) + \frac{d}{d x} \csc{\left(3 x \right)}}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
   sumApply the sum rule to the derivative of the inner expression.✓ Proved
4. \[ = \frac{- \frac{d}{d x} \cot{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
   constantFactor out the negative sign from the first term.✓ Proved
5. \[ = \frac{- 3 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + 3 \csc^{2}{\left(3 x \right)}}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
   trig algebraDifferentiate the trigonometric functions using the chain rule. Simplify the expression by distributing the negative signs.✓ Proved
6. \[ = 3 \csc{\left(3 x \right)} \]
   algebra simplifyFactor out the common term 3*csc(3*x) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved