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

Problem 2.514 · hard

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