Derivative of \( \displaystyle \frac{3 \ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} \)

Problem 2.665 · hard

1. \[ \frac{d}{d x} \frac{3 \ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
   logarithmicApply the derivative rule for the natural logarithm.✓ Proved
4. \[ = \frac{3 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(2 x - 3 \right)}\right)}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
   chainApply the chain rule to the inner function.✓ Proved
5. \[ = \frac{3 \frac{d}{d x} \tan^{2}{\left(2 x - 3 \right)}}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
   constant algebraThe derivative of the constant 1 is 0. Simplify the expression by removing the zero term.✓ Proved
6. \[ = \frac{3 \tan{\left(2 x - 3 \right)} \frac{d}{d x} \tan{\left(2 x - 3 \right)}}{2 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
   powerApply the power rule to the squared tangent term.✓ Proved
7. \[ = \frac{3 \tan{\left(2 x - 3 \right)} \sec^{2}{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
   trigThe derivative of tan(u) is sec(u)^2 * du/dx.≈ Checked numerically
8. \[ = \frac{3 \tan{\left(2 x - 3 \right)} \sec^{2}{\left(2 x - 3 \right)}}{\tan^{2}{\left(2 x - 3 \right)} + 1} \]
   chain algebra algebraDifferentiate the innermost linear function 2*x - 3. Multiply the constants together. Combine the terms into a single fraction.✓ Proved
9. \[ = 3 \tan{\left(2 x - 3 \right)} \]
   simplifyUse the identity 1 + tan(u)^2 = sec(u)^2 to simplify the fraction.≈ Checked numerically