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

Problem 2.289 · hard Beautiful

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