Derivative of \( \displaystyle \frac{\ln{\left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)} \right)}}{2} \)

Problem 2.352 · hard

1. \[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)} \right)}}{2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)} \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)}\right)}{2 \left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)}\right)} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = \frac{\frac{d}{d x} \tan{\left(2 x - 3 \right)} + \frac{d}{d x} \sec{\left(2 x - 3 \right)}}{2 \left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)}\right)} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = \frac{\tan{\left(2 x - 3 \right)} \sec{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right) + \sec^{2}{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)}\right)} \]
   trigApply the derivative rules for tan and sec.≈ Checked numerically
6. \[ = \frac{2 \tan{\left(2 x - 3 \right)} \sec{\left(2 x - 3 \right)} + 2 \sec^{2}{\left(2 x - 3 \right)}}{2 \left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)}\right)} \]
   chainApply the chain rule to the inner linear function.✓ Proved
7. \[ = \frac{\tan{\left(2 x - 3 \right)} \sec{\left(2 x - 3 \right)} + \sec^{2}{\left(2 x - 3 \right)}}{\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)}} \]
   algebra simplifyFactor out the common constant 2. Cancel the 1/2 and the 2.✓ Proved
8. \[ = \sec{\left(2 x - 3 \right)} \]
   algebra simplifyFactor out sec(2*x - 3) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved