Derivative of \( \displaystyle \frac{5 \ln{\left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)} \right)}}{3} \)

Problem 2.1514 · hard Beautiful

1. \[ \frac{d}{d x} \frac{5 \ln{\left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)} \right)}}{3} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{5 \frac{d}{d x} \ln{\left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)} \right)}}{3} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{5 \frac{d}{d x} \left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)}\right)}{3 \left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)}\right)} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = \frac{5 \left(\frac{d}{d x} \tan{\left(3 x \right)} + \frac{d}{d x} \sec{\left(3 x \right)}\right)}{3 \left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)}\right)} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = \frac{5 \left(3 \tan{\left(3 x \right)} \sec{\left(3 x \right)} + 3 \sec^{2}{\left(3 x \right)}\right)}{3 \left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)}\right)} \]
   chainApply the chain rule to the trigonometric terms.✓ Proved
6. \[ = \frac{5 \left(\tan{\left(3 x \right)} \sec{\left(3 x \right)} + \sec^{2}{\left(3 x \right)}\right)}{\tan{\left(3 x \right)} + \sec{\left(3 x \right)}} \]
   constant-multipleFactor out the constant 3.✓ Proved
7. \[ = \frac{5 \tan{\left(3 x \right)} \sec{\left(3 x \right)} + 5 \sec^{2}{\left(3 x \right)}}{\tan{\left(3 x \right)} + \sec{\left(3 x \right)}} \]
   algebraSimplify the constants and the expression.✓ Proved
8. \[ = 5 \sec{\left(3 x \right)} \]
   algebra simplifyFactor out sec(3*x) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved