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

Problem 2.400 · hard

1. \[ \frac{d}{d x} \operatorname{atan}{\left(\sin{\left(x - 1 \right)} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sin{\left(x - 1 \right)}}{\sin^{2}{\left(x - 1 \right)} + 1} \]
   chainApply the chain rule for the arctangent function.✓ Proved
3. \[ = \frac{\cos{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)}{\sin^{2}{\left(x - 1 \right)} + 1} \]
   trigApply the chain rule for the sine function.✓ Proved
4. \[ = \frac{\cos{\left(x - 1 \right)}}{\sin^{2}{\left(x - 1 \right)} + 1} \]
   derivative simplifyDifferentiate the inner linear function. Simplify the expression.✓ Proved