Derivative of \( \displaystyle \ln{\left(\frac{e^{1 - 2 x}}{\cos{\left(2 x - 1 \right)}} \right)} \)

Problem 2.1456 · hard

1. \[ \frac{d}{d x} \ln{\left(\frac{e^{1 - 2 x}}{\cos{\left(2 x - 1 \right)}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 2 x - \ln{\left(\cos{\left(2 x - 1 \right)} \right)} + 1\right) \]
   simplify simplifyUse logarithm properties to split the quotient. Simplify the log(exp(...)) term.✓ Proved
3. \[ = \frac{d}{d x} \left(1 - 2 x\right) + \frac{d}{d x} \left(- \ln{\left(\cos{\left(2 x - 1 \right)} \right)}\right) \]
   sumApply the sum rule for differentiation.✓ Proved
4. \[ = \frac{d}{d x} \left(1 - 2 x\right) - \frac{d}{d x} \ln{\left(\cos{\left(2 x - 1 \right)} \right)} \]
   constantFactor out the negative sign.✓ Proved
5. \[ = \frac{d}{d x} 1 - \frac{d}{d x} 2 x - \frac{d}{d x} \ln{\left(\cos{\left(2 x - 1 \right)} \right)} \]
   sumSplit the first term into two parts.✓ Proved
6. \[ = - \frac{d}{d x} 2 x - \frac{d}{d x} \ln{\left(\cos{\left(2 x - 1 \right)} \right)} \]
   constant algebraDifferentiate the constant term. Simplify the expression.✓ Proved
7. \[ = - \frac{d}{d x} \ln{\left(\cos{\left(2 x - 1 \right)} \right)} - 2 \]
   derivativeDifferentiate the 2*x term.✓ Proved
8. \[ = -2 - \frac{\frac{d}{d x} \cos{\left(2 x - 1 \right)}}{\cos{\left(2 x - 1 \right)}} \]
   chainApply the chain rule to the logarithm.✓ Proved
9. \[ = \frac{\sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{\cos{\left(2 x - 1 \right)}} - 2 \]
   chainApply the chain rule to the cosine term.✓ Proved
10. \[ = \frac{2 \sin{\left(2 x - 1 \right)}}{\cos{\left(2 x - 1 \right)}} - 2 \]
    derivative algebraDifferentiate the linear inner function. Simplify the signs and coefficients.✓ Proved
11. \[ = 2 \tan{\left(2 x - 1 \right)} - 2 \]
    simplifyUse the identity tan(u) = sin(u)/cos(u).✓ Proved