Derivative of \( \displaystyle \frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} - \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \)

Problem 2.1308 · hard

1. \[ \frac{d}{d x} \left(\frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} - \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} - \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \]
   constant-multiple algebraApply the constant multiple rule to each term. Factor out the common constant 1/4.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(\sin{\left(2 x \right)} + 1\right)}{4 \left(\sin{\left(2 x \right)} + 1\right)} + \frac{\frac{d}{d x} \left(\sin{\left(2 x \right)} - 1\right)}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
   chainApply the chain rule to the logarithmic terms.✓ Proved
4. \[ = - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \sin{\left(2 x \right)}}{4 \left(\sin{\left(2 x \right)} + 1\right)} + \frac{- \frac{d}{d x} 1 + \frac{d}{d x} \sin{\left(2 x \right)}}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
   sumDifferentiate the terms inside the parentheses.✓ Proved
5. \[ = - \frac{\frac{d}{d x} \sin{\left(2 x \right)}}{4 \left(\sin{\left(2 x \right)} + 1\right)} + \frac{\frac{d}{d x} \sin{\left(2 x \right)}}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
   constant algebraThe derivative of a constant is zero. Simplify the expression by removing zero terms.✓ Proved
6. \[ = \frac{\left(- \frac{1}{\sin{\left(2 x \right)} + 1} + \frac{1}{\sin{\left(2 x \right)} - 1}\right) \frac{d}{d x} \sin{\left(2 x \right)}}{4} \]
   algebraFactor out the common derivative term.✓ Proved
7. \[ = \frac{\left(- \frac{1}{\sin{\left(2 x \right)} + 1} + \frac{1}{\sin{\left(2 x \right)} - 1}\right) \cos{\left(2 x \right)}}{2} \]
   trig algebraDifferentiate the sine term using the chain rule. Simplify the constant coefficient.✓ Proved
8. \[ = \frac{\cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} - 1\right) \left(\sin{\left(2 x \right)} + 1\right)} \]
   algebraCombine the fractions using a common denominator.✓ Proved
9. \[ = \frac{\cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} - 1} \]
   algebra algebraSimplify the numerator and the denominator. Simplify the expression by canceling the 2 and the 1/2.✓ Proved
10. \[ = - \frac{\cos{\left(2 x \right)}}{1 - \sin^{2}{\left(2 x \right)}} \]
    algebraRewrite the denominator to use a trigonometric identity.✓ Proved
11. \[ = - \frac{1}{\cos{\left(2 x \right)}} \]
    rewrite simplifyUse the Pythagorean identity 1 - sin^2(u) = cos^2(u). Simplify the fraction by canceling cos(2*x).✓ Proved
12. \[ = - \sec{\left(2 x \right)} \]
    rewriteRewrite 1/cos(u) as sec(u).✓ Proved