Derivative of \( \displaystyle - 5 \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)} \)

Problem 2.474 · hard Beautiful

1. \[ \frac{d}{d x} \left(- 5 \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - 5 \frac{d}{d x} \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{5 \frac{d}{d x} \left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}\right)}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = - \frac{5 \left(\frac{d}{d x} \cot{\left(x - 1 \right)} + \frac{d}{d x} \csc{\left(x - 1 \right)}\right)}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = - \frac{5 \left(- \cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) - \csc^{2}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)\right)}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
   trigApply the derivative rules for cotangent and cosecant.✓ Proved
6. \[ = - \frac{5 \left(- \cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} - \csc^{2}{\left(x - 1 \right)}\right)}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
   derivative algebraDifferentiate the inner linear function x - 1. Simplify the expression.✓ Proved
7. \[ = \frac{5 \cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} + 5 \csc^{2}{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
   algebraDistribute the negative sign and the constant.✓ Proved
8. \[ = 5 \csc{\left(x - 1 \right)} \]
   algebra simplifyFactor out csc(x - 1) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved