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

Problem 2.339 · hard

1. \[ \frac{d}{d x} \frac{\sin{\left(2 x - 1 \right)}}{2 x - 1} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \sin{\left(2 x - 1 \right)} \frac{d}{d x} \frac{1}{2 x - 1} + \frac{\frac{d}{d x} \sin{\left(2 x - 1 \right)}}{2 x - 1} \]
   productApply the product rule.✓ Proved
3. \[ = \sin{\left(2 x - 1 \right)} \frac{d}{d x} \frac{1}{2 x - 1} + \frac{\cos{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]
   chainApply the chain rule to the first term.✓ Proved
4. \[ = \sin{\left(2 x - 1 \right)} \frac{d}{d x} \frac{1}{2 x - 1} + \frac{2 \cos{\left(2 x - 1 \right)}}{2 x - 1} \]
   derivativeDifferentiate the inner function 2*x - 1.✓ Proved
5. \[ = \frac{2 \cos{\left(2 x - 1 \right)}}{2 x - 1} - \frac{\sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{\left(2 x - 1\right)^{2}} \]
   chainApply the chain rule to the second term.✓ Proved
6. \[ = \frac{2 \cos{\left(2 x - 1 \right)}}{2 x - 1} - \frac{2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
   derivative algebraDifferentiate the inner function 2*x - 1. Simplify the terms using exponent rules.✓ Proved
7. \[ = \frac{2 \left(2 x - 1\right) \cos{\left(2 x - 1 \right)} - 2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved
8. \[ = \frac{\left(4 x - 2\right) \cos{\left(2 x - 1 \right)} - 2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
   simplifyFinal simplified form.✓ Proved