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

Problem 2.9 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(- \cot{\left(x \right)} + \csc{\left(x \right)} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(- \cot{\left(x \right)} + \csc{\left(x \right)}\right)}{- \cot{\left(x \right)} + \csc{\left(x \right)}} \]
   chainApply the chain rule for the logarithm.✓ Proved
3. \[ = \frac{- \frac{d}{d x} \cot{\left(x \right)} + \frac{d}{d x} \csc{\left(x \right)}}{- \cot{\left(x \right)} + \csc{\left(x \right)}} \]
   sumDistribute the derivative over the sum.✓ Proved
4. \[ = \frac{- \cot{\left(x \right)} \csc{\left(x \right)} + \csc^{2}{\left(x \right)}}{- \cot{\left(x \right)} + \csc{\left(x \right)}} \]
   trig algebraApply the derivatives of cot(x) and csc(x). Simplify the signs inside the parentheses.✓ Proved
5. \[ = \csc{\left(x \right)} \]
   algebra simplifyFactor out csc(x) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved