Derivative of \( \displaystyle - \frac{\ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} + \frac{\ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \)

Problem 2.245 · hard

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} + \frac{\ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8}\right) \]
   sumApply the sum rule for differentiation.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8}\right) + \frac{d}{d x} \frac{\ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \]
   constant-multipleSeparate the terms and pull out the constant factor 1/8.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} + \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \]
   constant-multiplePull out the negative sign from the first term.✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(\cos{\left(4 x \right)} + 1\right)}{8 \left(\cos{\left(4 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\cos{\left(4 x \right)} - 1\right)}{8 \left(\cos{\left(4 x \right)} - 1\right)} \]
   chainApply the chain rule to the logarithmic terms.✓ Proved
5. \[ = - \frac{\sin{\left(4 x \right)}}{2 \left(\cos{\left(4 x \right)} + 1\right)} + \frac{\sin{\left(4 x \right)}}{2 \left(\cos{\left(4 x \right)} - 1\right)} \]
   chainDifferentiate the inner functions using the chain rule.✓ Proved
6. \[ = - \frac{4 \sin{\left(4 x \right)}}{8 \cos{\left(4 x \right)} + 8} + \frac{4 \sin{\left(4 x \right)}}{8 \cos{\left(4 x \right)} - 8} \]
   algebraSimplify the products of constants and signs.✓ Proved
7. \[ = - \frac{\sin{\left(4 x \right)}}{2 \cos{\left(4 x \right)} + 2} + \frac{\sin{\left(4 x \right)}}{2 \cos{\left(4 x \right)} - 2} \]
   simplifySimplify the fraction 4/8 to 1/2.✓ Proved
8. \[ = \frac{- \left(\cos{\left(4 x \right)} - 1\right) \sin{\left(4 x \right)} + \left(\cos{\left(4 x \right)} + 1\right) \sin{\left(4 x \right)}}{\left(\cos{\left(4 x \right)} + 1\right) \left(2 \cos{\left(4 x \right)} - 2\right)} \]
   algebraFind a common denominator to combine the fractions.✓ Proved
9. \[ = \frac{2 \sin{\left(4 x \right)}}{2 \cos^{2}{\left(4 x \right)} - 2} \]
   algebra simplifyDistribute the terms in the numerator and denominator. Simplify the numerator by combining like terms.✓ Proved
10. \[ = \frac{\sin{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)} - 1} \]
    simplifyCancel the common factor of 2.✓ Proved
11. \[ = - \frac{\sin{\left(4 x \right)}}{1 - \cos^{2}{\left(4 x \right)}} \]
    rewriteRewrite the denominator using the Pythagorean identity.✓ Proved
12. \[ = - \frac{1}{\sin{\left(4 x \right)}} \]
    rewrite simplifyUse the identity 1 - cos(x)^2 = sin(x)^2. Cancel one sin(4*x) from the numerator and denominator.✓ Proved
13. \[ = - \csc{\left(4 x \right)} \]
    simplifyRewrite 1/sin(x) as csc(x).✓ Proved