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

Problem 2.568 · hard

1. \[ \frac{d}{d x} \frac{\cos{\left(2 x + 2 \right)}}{2 x + 2} \]
   derivative quotientStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \cos{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{2 x + 2} + \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} \]
   productApply the product rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} - \frac{\cos{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{\left(2 x + 2\right)^{2}} \]
   chainApply the chain rule to the second term.✓ Proved
4. \[ = \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} - \frac{2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
   derivativeDifferentiate the inner function 2*x + 2.✓ Proved
5. \[ = - \frac{2 \sin{\left(2 x + 2 \right)}}{2 x + 2} - \frac{2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
   chain algebra algebraApply the chain rule to the first term. Distribute and simplify the terms. Rewrite negative exponents as denominators.✓ Proved
6. \[ = \frac{- 2 \left(2 x + 2\right) \sin{\left(2 x + 2 \right)} - 2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
   algebra simplifyFind a common denominator. Factor out the common term -2.✓ Proved