Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(3 x \right)} - 1 \right)}}{6} - \frac{\ln{\left(\cos{\left(3 x \right)} + 1 \right)}}{6} \)

Problem 2.901 · hard Beautiful

1. \[ \frac{d}{d x} \left(\frac{\ln{\left(\cos{\left(3 x \right)} - 1 \right)}}{6} - \frac{\ln{\left(\cos{\left(3 x \right)} + 1 \right)}}{6}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{\ln{\left(\cos{\left(3 x \right)} - 1 \right)}}{6} - \frac{d}{d x} \frac{\ln{\left(\cos{\left(3 x \right)} + 1 \right)}}{6} \]
   sumApply the difference rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(3 x \right)} - 1 \right)}}{6} - \frac{\frac{d}{d x} \ln{\left(\cos{\left(3 x \right)} + 1 \right)}}{6} \]
   constant-multipleFactor out the constant 1/6.✓ Proved
4. \[ = - \frac{\frac{d}{d x} \left(\cos{\left(3 x \right)} + 1\right)}{6 \left(\cos{\left(3 x \right)} + 1\right)} + \frac{\frac{d}{d x} \left(\cos{\left(3 x \right)} - 1\right)}{6 \left(\cos{\left(3 x \right)} - 1\right)} \]
   chainApply the chain rule to each logarithmic term.✓ Proved
5. \[ = - \frac{\frac{d}{d x} \cos{\left(3 x \right)}}{6 \left(\cos{\left(3 x \right)} + 1\right)} + \frac{\frac{d}{d x} \cos{\left(3 x \right)}}{6 \left(\cos{\left(3 x \right)} - 1\right)} \]
   derivative derivativeDifferentiate the inner terms. Simplify the derivative of the constants.✓ Proved
6. \[ = \frac{\sin{\left(3 x \right)}}{2 \left(\cos{\left(3 x \right)} + 1\right)} - \frac{\sin{\left(3 x \right)}}{2 \left(\cos{\left(3 x \right)} - 1\right)} \]
   chain algebra algebraDifferentiate the cosine terms. Multiply the terms together. Simplify the coefficients.✓ Proved
7. \[ = \frac{\left(\frac{1}{\cos{\left(3 x \right)} + 1} - \frac{1}{\cos{\left(3 x \right)} - 1}\right) \sin{\left(3 x \right)}}{2} \]
   algebraFactor out the common term sin(3*x)/2.✓ Proved
8. \[ = - \frac{\sin{\left(3 x \right)}}{\left(\cos{\left(3 x \right)} - 1\right) \left(\cos{\left(3 x \right)} + 1\right)} \]
   algebraFind a common denominator.✓ Proved
9. \[ = - \frac{\sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)} - 1} \]
   algebra algebraSimplify the numerator and denominator. Cancel the common factor of 2.✓ Proved
10. \[ = \frac{\sin{\left(3 x \right)}}{1 - \cos^{2}{\left(3 x \right)}} \]
    algebraDistribute the negative sign.✓ Proved
11. \[ = \frac{1}{\sin{\left(3 x \right)}} \]
    algebra simplifyUse the identity 1 - cos(2u) = sin(2u) logic, or specifically 1 - cos(a)^2 = sin(a)^2. Cancel sin(3*x) from numerator and denominator.✓ Proved
12. \[ = \csc{\left(3 x \right)} \]
    rewriteRewrite in terms of cosecant.✓ Proved