Derivative of \( \displaystyle - \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x - 1 \right)} \right)}}{2} \)

Problem 2.112 · hard

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x - 1 \right)} \right)}}{2}\right) \]
   derivative constant-multipleStart with the derivative of the function. Distribute the constant factors.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4}\right) + \frac{d}{d x} \frac{\ln{\left(\tan{\left(2 x - 1 \right)} \right)}}{2} \]
   sumApply the sum rule for derivatives.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4}\right) + \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x - 1 \right)} \right)}}{2} \]
   constant-multipleFactor out the constant from the second term.✓ Proved
4. \[ = \frac{\frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \tan{\left(2 x - 1 \right)}} - \frac{\frac{d}{d x} \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)}{4 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]
   chainApply the chain rule to both logarithmic terms.✓ Proved
5. \[ = \frac{\frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \tan{\left(2 x - 1 \right)}} - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(2 x - 1 \right)}}{4 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]
   sumApply the sum rule to the inner derivative.✓ Proved
6. \[ = \frac{\frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \tan{\left(2 x - 1 \right)}} - \frac{\tan{\left(2 x - 1 \right)} \frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]
   powerApply the power rule to the squared tangent term.✓ Proved
7. \[ = \frac{\sec^{2}{\left(2 x - 1 \right)}}{\tan{\left(2 x - 1 \right)}} - \frac{\tan{\left(2 x - 1 \right)} \sec^{2}{\left(2 x - 1 \right)}}{\tan^{2}{\left(2 x - 1 \right)} + 1} \]
   chain algebra algebraApply the chain rule to the derivative of tan(2*x - 1). Simplify the products in the numerators. Simplify the fractions by combining constants.≈ Checked numerically
8. \[ = - \tan{\left(2 x - 1 \right)} + \frac{\sec^{2}{\left(2 x - 1 \right)}}{\tan{\left(2 x - 1 \right)}} \]
   rewrite simplifyUse the identity tan(u)**2 + 1 = sec(u)**2. Cancel the sec(2*x - 1)**2 term in the first fraction.≈ Checked numerically
9. \[ = \frac{\tan^{2}{\left(2 x - 1 \right)} + 1}{\tan{\left(2 x - 1 \right)}} - \tan{\left(2 x - 1 \right)} \]
   rewriteRewrite sec(2*x - 1)**2 in terms of tangent.≈ Checked numerically
10. \[ = \frac{1}{\tan{\left(2 x - 1 \right)}} \]
    algebra simplifySplit the fraction into two parts. Combine the tangent terms.✓ Proved
11. \[ = \cot{\left(2 x - 1 \right)} \]
    simplifyUse the definition of the cotangent function.✓ Proved