Derivative of \( \displaystyle - \frac{5 \ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} + \frac{5 \ln{\left(\tan{\left(2 x - 1 \right)} \right)}}{2} \)

Problem 2.1398 · hard

1. \[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} + \frac{5 \ln{\left(\tan{\left(2 x - 1 \right)} \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{5 \frac{d}{d x} \ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} + \frac{5 \frac{d}{d x} \ln{\left(\tan{\left(2 x - 1 \right)} \right)}}{2} \]
   algebraDistribute the constant and split the fraction.✓ Proved
3. \[ = \frac{5 \frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \tan{\left(2 x - 1 \right)}} - \frac{5 \frac{d}{d x} \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)}{4 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]
   chainApply the chain rule to the logarithmic terms.✓ Proved
4. \[ = \frac{5 \frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \tan{\left(2 x - 1 \right)}} - \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(2 x - 1 \right)}\right)}{4 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]
   sumApply the sum rule to the inner derivative.✓ Proved
5. \[ = \frac{5 \frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \tan{\left(2 x - 1 \right)}} - \frac{5 \frac{d}{d x} \tan^{2}{\left(2 x - 1 \right)}}{4 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = \frac{5 \frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \tan{\left(2 x - 1 \right)}} - \frac{5 \tan{\left(2 x - 1 \right)} \frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]
   powerApply the power rule to the squared tangent term.✓ Proved
7. \[ = \frac{5 \sec^{2}{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{2 \tan{\left(2 x - 1 \right)}} - \frac{5 \tan{\left(2 x - 1 \right)} \sec^{2}{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{2 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]
   chainApply the chain rule to the tangent function.≈ Checked numerically
8. \[ = \frac{5 \sec^{2}{\left(2 x - 1 \right)}}{\tan{\left(2 x - 1 \right)}} - \frac{5 \tan{\left(2 x - 1 \right)} \sec^{2}{\left(2 x - 1 \right)}}{\tan^{2}{\left(2 x - 1 \right)} + 1} \]
   constant algebra algebraDifferentiate the linear term 2x - 1. Simplify the products in each term. Simplify the coefficients and combine terms.✓ Proved
9. \[ = 5 \left(\frac{1}{\tan{\left(2 x - 1 \right)}} - \frac{\tan{\left(2 x - 1 \right)}}{\tan^{2}{\left(2 x - 1 \right)} + 1}\right) \sec^{2}{\left(2 x - 1 \right)} \]
   algebraFactor out the common term 5*sec(2*x - 1)**2.✓ Proved
10. \[ = 5 \left(- \frac{\tan{\left(2 x - 1 \right)}}{\sec^{2}{\left(2 x - 1 \right)}} + \frac{1}{\tan{\left(2 x - 1 \right)}}\right) \sec^{2}{\left(2 x - 1 \right)} \]
    algebraUse the identity 1 + tan(u)**2 = sec(u)**2.≈ Checked numerically
11. \[ = 5 \left(- \cos^{2}{\left(2 x - 1 \right)} \tan{\left(2 x - 1 \right)} + \frac{1}{\tan{\left(2 x - 1 \right)}}\right) \sec^{2}{\left(2 x - 1 \right)} \]
    algebraRewrite 1/sec(u)**2 as cos(u)**2.✓ Proved
12. \[ = 5 \left(- \cos^{2}{\left(2 x - 1 \right)} \tan{\left(2 x - 1 \right)} + \cot{\left(2 x - 1 \right)}\right) \sec^{2}{\left(2 x - 1 \right)} \]
    algebraConvert 1/tan(u) to cot(u).✓ Proved
13. \[ = 5 \left(- \sin{\left(2 x - 1 \right)} \cos{\left(2 x - 1 \right)} + \cot{\left(2 x - 1 \right)}\right) \sec^{2}{\left(2 x - 1 \right)} \]
    algebra algebraConvert tan(u) to sin(u)/cos(u). Simplify the product of sin and cos.✓ Proved
14. \[ = 5 \left(- \sin{\left(2 x - 1 \right)} \cos{\left(2 x - 1 \right)} + \frac{\cos{\left(2 x - 1 \right)}}{\sin{\left(2 x - 1 \right)}}\right) \sec^{2}{\left(2 x - 1 \right)} \]
    algebraConvert cot(u) to cos(u)/sin(u).✓ Proved
15. \[ = 5 \left(- \sin{\left(2 x - 1 \right)} + \frac{1}{\sin{\left(2 x - 1 \right)}}\right) \cos{\left(2 x - 1 \right)} \sec^{2}{\left(2 x - 1 \right)} \]
    algebraFactor out cos(2*x - 1).✓ Proved
16. \[ = \frac{5 \left(1 - \sin^{2}{\left(2 x - 1 \right)}\right) \cos{\left(2 x - 1 \right)} \sec^{2}{\left(2 x - 1 \right)}}{\sin{\left(2 x - 1 \right)}} \]
    algebraFind a common denominator inside the parenthesis.✓ Proved
17. \[ = \frac{5 \cos^{3}{\left(2 x - 1 \right)} \sec^{2}{\left(2 x - 1 \right)}}{\sin{\left(2 x - 1 \right)}} \]
    algebra algebraUse the identity 1 - sin(u)**2 = cos(u)**2. Combine the cosine terms.✓ Proved
18. \[ = \frac{5 \cos{\left(2 x - 1 \right)}}{\sin{\left(2 x - 1 \right)}} \]
    algebra simplifyRewrite sec(u)**2 as 1/cos(u)**2. Simplify the expression by canceling cos(2*x - 1)**2.✓ Proved