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

Problem 2.312 · hard

1. \[ \frac{d}{d x} \frac{\sin{\left(3 x - 3 \right)}}{3 x - 2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sin{\left(3 x - 3 \right)}}{3 x - 2} - \frac{\sin{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 2\right)}{\left(3 x - 2\right)^{2}} \]
   quotientApply the quotient rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} \sin{\left(3 x - 3 \right)}}{3 x - 2} - \frac{\sin{\left(3 x - 3 \right)} \frac{d}{d x} 3 x}{\left(3 x - 2\right)^{2}} \]
   constantSeparate the derivative of the denominator.✓ Proved
4. \[ = \frac{\frac{d}{d x} \sin{\left(3 x - 3 \right)}}{3 x - 2} - \frac{3 \sin{\left(3 x - 3 \right)}}{\left(3 x - 2\right)^{2}} \]
   derivativeDifferentiate 3*x.✓ Proved
5. \[ = \frac{\cos{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)}{3 x - 2} - \frac{3 \sin{\left(3 x - 3 \right)}}{\left(3 x - 2\right)^{2}} \]
   chainApply the chain rule to the sine term.✓ Proved
6. \[ = \frac{3 \cos{\left(3 x - 3 \right)}}{3 x - 2} - \frac{3 \sin{\left(3 x - 3 \right)}}{\left(3 x - 2\right)^{2}} \]
   derivative constant-multipleDifferentiate 3*x - 3. Move the constant 3 to the numerator.✓ Proved
7. \[ = \frac{3 \left(3 x - 2\right) \cos{\left(3 x - 3 \right)} - 3 \sin{\left(3 x - 3 \right)}}{\left(3 x - 2\right)^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved