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

Problem 2.555 · hard Beautiful

1. \[ \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(2 x + 1 \right)} + 1 \right)}}{4} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(2 x + 1 \right)} + 1 \right)}}{4} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\tan^{2}{\left(2 x + 1 \right)} + 1\right)}{4 \left(\tan^{2}{\left(2 x + 1 \right)} + 1\right)} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(2 x + 1 \right)}}{4 \left(\tan^{2}{\left(2 x + 1 \right)} + 1\right)} \]
   sumApply the sum rule.✓ Proved
5. \[ = \frac{\frac{d}{d x} \tan^{2}{\left(2 x + 1 \right)}}{4 \left(\tan^{2}{\left(2 x + 1 \right)} + 1\right)} \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = \frac{\tan{\left(2 x + 1 \right)} \frac{d}{d x} \tan{\left(2 x + 1 \right)}}{2 \left(\tan^{2}{\left(2 x + 1 \right)} + 1\right)} \]
   powerApply the power rule.✓ Proved
7. \[ = \frac{\tan{\left(2 x + 1 \right)} \sec^{2}{\left(2 x + 1 \right)}}{\tan^{2}{\left(2 x + 1 \right)} + 1} \]
   chain algebraApply the chain rule to the tangent function. Simplify the constants.≈ Checked numerically
8. \[ = \tan{\left(2 x + 1 \right)} \]
   trig algebra simplifyUse the identity sec(u)**2 = 1 + tan(u)**2. Multiply the terms together. Cancel the common terms.≈ Checked numerically