Derivative of \( \displaystyle \frac{3 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \)

Problem 2.63 · hard

1. \[ \frac{d}{d x} \frac{3 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \]
   constant-multiplePull out the constant factor 3/2.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \sin{\left(2 x \right)}}{2 \sin{\left(2 x \right)}} \]
   chainApply the chain rule to the logarithm.✓ Proved
4. \[ = \frac{3 \cos{\left(2 x \right)} \frac{d}{d x} 2 x}{2 \sin{\left(2 x \right)}} \]
   chainApply the chain rule to the sine function.✓ Proved
5. \[ = \frac{3 \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} \]
   derivative algebra algebraDifferentiate the innermost function 2*x. Simplify the constant multiplication. Combine the terms into a single fraction.✓ Proved
6. \[ = 3 \cot{\left(2 x \right)} \]
   simplifyUse the trigonometric identity cot(u) = cos(u)/sin(u).✓ Proved