Derivative of \( \displaystyle \ln{\left(\tan{\left(5 x + 2 \right)} + \sec{\left(5 x + 2 \right)} \right)} \)

Problem 2.1076 · hard

1. \[ \frac{d}{d x} \ln{\left(\tan{\left(5 x + 2 \right)} + \sec{\left(5 x + 2 \right)} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\tan{\left(5 x + 2 \right)} + \sec{\left(5 x + 2 \right)}\right)}{\tan{\left(5 x + 2 \right)} + \sec{\left(5 x + 2 \right)}} \]
   chainApply the chain rule for the natural logarithm.✓ Proved
3. \[ = \frac{\frac{d}{d x} \tan{\left(5 x + 2 \right)} + \frac{d}{d x} \sec{\left(5 x + 2 \right)}}{\tan{\left(5 x + 2 \right)} + \sec{\left(5 x + 2 \right)}} \]
   sumApply the sum rule to the inner terms.✓ Proved
4. \[ = \frac{5 \tan{\left(5 x + 2 \right)} \sec{\left(5 x + 2 \right)} + 5 \sec^{2}{\left(5 x + 2 \right)}}{\tan{\left(5 x + 2 \right)} + \sec{\left(5 x + 2 \right)}} \]
   chainDifferentiate the tangent and secant terms using the chain rule.≈ Checked numerically
5. \[ = \frac{5 \left(\frac{\tan{\left(5 x + 2 \right)}}{\sec{\left(5 x + 2 \right)}} + 1\right) \sec^{2}{\left(5 x + 2 \right)}}{\tan{\left(5 x + 2 \right)} + \sec{\left(5 x + 2 \right)}} \]
   algebraFactor out common terms from the numerator.✓ Proved
6. \[ = \frac{5 \tan{\left(5 x + 2 \right)} \sec{\left(5 x + 2 \right)} + 5 \sec^{2}{\left(5 x + 2 \right)}}{\tan{\left(5 x + 2 \right)} + \sec{\left(5 x + 2 \right)}} \]
   algebraDistribute the constant factor.✓ Proved
7. \[ = 5 \sec{\left(5 x + 2 \right)} \]
   algebra simplifyFactor out sec(5*x + 2) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved