Derivative of \( \displaystyle \frac{\ln{\left(\tan^{2}{\left(5 x + 2 \right)} + 1 \right)}}{10} \)

Problem 2.980 · hard Beautiful

1. \[ \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(5 x + 2 \right)} + 1 \right)}}{10} \]
   constant-multiplePull out the constant factor.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(5 x + 2 \right)} + 1 \right)}}{10} \]
   chainApply the chain rule to the logarithm.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\tan^{2}{\left(5 x + 2 \right)} + 1\right)}{10 \left(\tan^{2}{\left(5 x + 2 \right)} + 1\right)} \]
   sumDifferentiate the sum inside the derivative.✓ Proved
4. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(5 x + 2 \right)}}{10 \left(\tan^{2}{\left(5 x + 2 \right)} + 1\right)} \]
   constantThe derivative of a constant is zero.✓ Proved
5. \[ = \frac{\frac{d}{d x} \tan^{2}{\left(5 x + 2 \right)}}{10 \left(\tan^{2}{\left(5 x + 2 \right)} + 1\right)} \]
   powerApply the power rule to the squared tangent.✓ Proved
6. \[ = \frac{\tan{\left(5 x + 2 \right)} \frac{d}{d x} \tan{\left(5 x + 2 \right)}}{5 \left(\tan^{2}{\left(5 x + 2 \right)} + 1\right)} \]
   chainApply the chain rule to the tangent function.✓ Proved
7. \[ = \frac{\tan{\left(5 x + 2 \right)} \sec^{2}{\left(5 x + 2 \right)} \frac{d}{d x} \left(5 x + 2\right)}{5 \left(\tan^{2}{\left(5 x + 2 \right)} + 1\right)} \]
   derivativeDifferentiate the linear inner function.≈ Checked numerically
8. \[ = \frac{\tan{\left(5 x + 2 \right)} \sec^{2}{\left(5 x + 2 \right)}}{\tan^{2}{\left(5 x + 2 \right)} + 1} \]
   algebra algebra simplify simplifySimplify the constants. Multiply the constants together. Cancel the 10s and simplify the expression. Use the identity sec(u)**2 = 1 + tan(u)**2 to simplify the denominator.✓ Proved
9. \[ = \tan{\left(5 x + 2 \right)} \]
   simplify simplifyCancel the sec(5*x + 2)**2 term from numerator and denominator. Final simplified result.≈ Checked numerically