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

Problem 2.906 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{e^{1 - x}}{\cos{\left(x - 1 \right)}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x - \ln{\left(\cos{\left(x - 1 \right)} \right)} + 1\right) \]
   algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(1 - x)) to 1 - x.✓ Proved
3. \[ = \frac{d}{d x} \left(1 - x\right) - \frac{d}{d x} \ln{\left(\cos{\left(x - 1 \right)} \right)} \]
   sumApply the linearity of the derivative.✓ Proved
4. \[ = - \frac{d}{d x} \ln{\left(\cos{\left(x - 1 \right)} \right)} - 1 \]
   constantThe derivative of 1 - x is -1.✓ Proved
5. \[ = -1 - \frac{\frac{d}{d x} \cos{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} \]
   chainApply the chain rule to the second term.✓ Proved
6. \[ = \frac{\sin{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} - 1 \]
   trig algebraThe derivative of cos(x - 1) is -sin(x - 1). Simplify the fraction.✓ Proved
7. \[ = \tan{\left(x - 1 \right)} - 1 \]
   trig algebraUse the identity tan(u) = sin(u)/cos(u). Rearrange the terms.✓ Proved