Derivative of \( \displaystyle \operatorname{atan}{\left(\cot{\left(2 x - 1 \right)} \right)} \)

Problem 2.535 · hard Beautiful

1. \[ \frac{d}{d x} \operatorname{atan}{\left(\cot{\left(2 x - 1 \right)} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \cot{\left(2 x - 1 \right)}}{\cot^{2}{\left(2 x - 1 \right)} + 1} \]
   chainApply the chain rule to the arctangent function.✓ Proved
3. \[ = - \frac{\csc^{2}{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{\cot^{2}{\left(2 x - 1 \right)} + 1} \]
   chainApply the chain rule to the cotangent function.✓ Proved
4. \[ = - \frac{2 \csc^{2}{\left(2 x - 1 \right)}}{\cot^{2}{\left(2 x - 1 \right)} + 1} \]
   derivative algebraDifferentiate the innermost linear function. Multiply the terms together.✓ Proved
5. \[ = -2 \]
   algebra simplifyUse the identity 1 + cot(u)**2 = csc(u)**2. Simplify the resulting expression.✓ Proved