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

Problem 2.322 · hard

1. \[ \frac{d}{d x} \frac{\sin{\left(2 x + 1 \right)}}{2 x + 1} \]
   rewriteStart with the derivative of the function. Rewrite the denominator 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. \[ = \sin{\left(2 x + 1 \right)} \frac{d}{d x} \frac{1}{2 x + 1} + \frac{2 \cos{\left(2 x + 1 \right)}}{2 x + 1} \]
   chain constant-multipleDifferentiate the first term using the chain rule. Factor out the constant 2.✓ Proved
4. \[ = \frac{2 \cos{\left(2 x + 1 \right)}}{2 x + 1} - \frac{\sin{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x + 1\right)}{\left(2 x + 1\right)^{2}} \]
   chainDifferentiate the second term using the chain rule.✓ Proved
5. \[ = \frac{2 \cos{\left(2 x + 1 \right)}}{2 x + 1} - \frac{2 \sin{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \]
   derivative algebra algebraDifferentiate the inner function 2*x + 1. Simplify the expression by multiplying the constants. Rewrite the negative exponents as denominators.✓ Proved
6. \[ = \frac{2 \left(2 x + 1\right) \cos{\left(2 x + 1 \right)} - 2 \sin{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \]
   algebra algebraCombine the terms over a common denominator. Factor out the common factor 2.✓ Proved