Derivative of \( \displaystyle \frac{\ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} \)

Problem 2.662 · hard Beautiful

1. \[ \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} \]
   constant-multiplePull out the constant factor 1/2.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\tan^{2}{\left(x - 1 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = \frac{\frac{d}{d x} \tan^{2}{\left(x - 1 \right)}}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{\tan{\left(x - 1 \right)} \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
   powerApply the power rule to the squared tangent term.✓ Proved
6. \[ = \frac{\tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)}{\tan^{2}{\left(x - 1 \right)} + 1} \]
   chainApply the chain rule to the tangent function.≈ Checked numerically
7. \[ = \frac{\tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
   derivative simplifyDifferentiate the inner linear function. Cancel the 1/2 and the 2.✓ Proved
8. \[ = \tan{\left(x - 1 \right)} \]
   algebra simplifyUse the identity tan(x - 1)**2 + 1 = sec(x - 1)**2. Simplify the fraction by canceling sec(x - 1)**2.≈ Checked numerically