Derivative of \( \displaystyle \frac{\ln{\left(\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)} \right)}}{4} \)

Problem 2.1029 · hard

1. \[ \frac{d}{d x} \frac{\ln{\left(\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)} \right)}}{4} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)} \right)}}{4} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)}\right)}{4 \left(\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)}\right)} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = \frac{\frac{d}{d x} \cot{\left(4 x + 1 \right)} + \frac{d}{d x} \csc{\left(4 x + 1 \right)}}{4 \left(\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)}\right)} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = \frac{- \cot{\left(4 x + 1 \right)} \csc{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 1\right) - \csc^{2}{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 1\right)}{4 \left(\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)}\right)} \]
   trigApply the derivatives of cot and csc.✓ Proved
6. \[ = \frac{- 4 \cot{\left(4 x + 1 \right)} \csc{\left(4 x + 1 \right)} - 4 \csc^{2}{\left(4 x + 1 \right)}}{4 \left(\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)}\right)} \]
   chainDifferentiate the inner linear function 4x + 1.✓ Proved
7. \[ = \frac{- \cot{\left(4 x + 1 \right)} \csc{\left(4 x + 1 \right)} - \csc^{2}{\left(4 x + 1 \right)}}{\cot{\left(4 x + 1 \right)} + \csc{\left(4 x + 1 \right)}} \]
   constant-multiple simplifyFactor out the 4. Cancel the 4 and the 1/4.✓ Proved
8. \[ = - \csc{\left(4 x + 1 \right)} \]
   algebra simplifyFactor out -csc(4*x + 1) from the numerator. Simplify the fraction.✓ Proved