Derivative of \( \displaystyle \ln{\left(\frac{e^{x + 1}}{\cos{\left(x + 1 \right)}} \right)} \)

Problem 2.531 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{e^{x + 1}}{\cos{\left(x + 1 \right)}} \right)} \]
   derivative algebraStart with the derivative of the function. Rewrite the quotient as a product.✓ Proved
2. \[ = \frac{d}{d x} \left(x + \ln{\left(\frac{1}{\cos{\left(x + 1 \right)}} \right)} + 1\right) \]
   logarithmic logarithmicApply the logarithm product rule. Simplify log(exp(x + 1)).✓ Proved
3. \[ = \frac{d}{d x} \left(x - \ln{\left(\cos{\left(x + 1 \right)} \right)} + 1\right) \]
   logarithmic constantApply the logarithm quotient rule. Simplify log(1) to 0.✓ Proved
4. \[ = \frac{d}{d x} \left(x + 1\right) - \frac{d}{d x} \ln{\left(\cos{\left(x + 1 \right)} \right)} \]
   sumApply the difference rule for derivatives.✓ Proved
5. \[ = 1 - \frac{d}{d x} \ln{\left(\cos{\left(x + 1 \right)} \right)} \]
   derivativeDifferentiate the first term.✓ Proved
6. \[ = 1 - \frac{\frac{d}{d x} \cos{\left(x + 1 \right)}}{\cos{\left(x + 1 \right)}} \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = \frac{\sin{\left(x + 1 \right)}}{\cos{\left(x + 1 \right)}} + 1 \]
   trig algebraDifferentiate the cosine function. Simplify the signs and the fraction.✓ Proved
8. \[ = \tan{\left(x + 1 \right)} + 1 \]
   trigUse the identity tan(u) = sin(u)/cos(u).✓ Proved