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

Problem 2.628 · hard

1. \[ \frac{d}{d x} \frac{\sin{\left(2 x - 3 \right)}}{2 x - 3} \]
   rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
2. \[ = \sin{\left(2 x - 3 \right)} \frac{d}{d x} \frac{1}{2 x - 3} + \frac{\frac{d}{d x} \sin{\left(2 x - 3 \right)}}{2 x - 3} \]
   productApply the product rule.✓ Proved
3. \[ = \sin{\left(2 x - 3 \right)} \frac{d}{d x} \frac{1}{2 x - 3} + \frac{2 \cos{\left(2 x - 3 \right)}}{2 x - 3} \]
   chainDifferentiate the first term using the chain rule.✓ Proved
4. \[ = \frac{2 \cos{\left(2 x - 3 \right)}}{2 x - 3} - \frac{2 \sin{\left(2 x - 3 \right)}}{\left(2 x - 3\right)^{2}} \]
   chain algebraDifferentiate the second term using the chain rule. Simplify the expression by combining terms.✓ Proved
5. \[ = \frac{2 \left(2 x - 3\right) \cos{\left(2 x - 3 \right)} - 2 \sin{\left(2 x - 3 \right)}}{\left(2 x - 3\right)^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved
6. \[ = \frac{\left(4 x - 6\right) \cos{\left(2 x - 3 \right)} - 2 \sin{\left(2 x - 3 \right)}}{\left(2 x - 3\right)^{2}} \]
   simplifyFinal simplified form.✓ Proved