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

Problem 2.1449 · hard Beautiful

1. \[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} + 5 \ln{\left(\tan{\left(x + 2 \right)} \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2}\right) + \frac{d}{d x} 5 \ln{\left(\tan{\left(x + 2 \right)} \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2}\right) + 5 \frac{d}{d x} \ln{\left(\tan{\left(x + 2 \right)} \right)} \]
   constant-multipleFactor out the constant from the second term.✓ Proved
4. \[ = \frac{5 \frac{d}{d x} \tan{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} - \frac{5 \frac{d}{d x} \left(\tan^{2}{\left(x + 2 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]
   logarithmicApply the chain rule for the logarithm.✓ Proved
5. \[ = \frac{5 \frac{d}{d x} \tan{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} - \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x + 2 \right)}\right)}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]
   sumApply the sum rule to the inner derivative.✓ Proved
6. \[ = \frac{5 \frac{d}{d x} \tan{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} - \frac{5 \frac{d}{d x} \tan^{2}{\left(x + 2 \right)}}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]
   constantThe derivative of a constant is zero.✓ Proved
7. \[ = \frac{5 \frac{d}{d x} \tan{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} - \frac{5 \tan{\left(x + 2 \right)} \frac{d}{d x} \tan{\left(x + 2 \right)}}{\tan^{2}{\left(x + 2 \right)} + 1} \]
   powerApply the power rule to the squared term.✓ Proved
8. \[ = \frac{5 \sec^{2}{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} - \frac{5 \tan{\left(x + 2 \right)} \sec^{2}{\left(x + 2 \right)}}{\tan^{2}{\left(x + 2 \right)} + 1} \]
   trig algebraSubstitute the derivative of tan(x+2). Simplify the coefficients.≈ Checked numerically
9. \[ = \frac{5 \left(\tan^{2}{\left(x + 2 \right)} + 1\right) \sec^{2}{\left(x + 2 \right)} - 5 \tan^{2}{\left(x + 2 \right)} \sec^{2}{\left(x + 2 \right)}}{\left(\tan^{2}{\left(x + 2 \right)} + 1\right) \tan{\left(x + 2 \right)}} \]
   algebraFind a common denominator.✓ Proved
10. \[ = \frac{5 \sec^{2}{\left(x + 2 \right)}}{\left(\tan^{2}{\left(x + 2 \right)} + 1\right) \tan{\left(x + 2 \right)}} \]
    algebra simplifyDistribute the numerator. Cancel the common terms in the numerator.✓ Proved
11. \[ = \frac{5}{\tan{\left(x + 2 \right)}} \]
    algebra simplifyUse the identity 1 + tan(x)^2 = sec(x)^2. Cancel the sec(x+2)^2 term.≈ Checked numerically
12. \[ = 5 \cot{\left(x + 2 \right)} \]
    trigUse the identity 1/tan(x) = cot(x).✓ Proved