Derivative of \( \displaystyle - \frac{\sqrt{2} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{6} \)

Problem 2.1136 · hard

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{6}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{6} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{\sqrt{2} \left(e^{3 x + 2} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} e^{3 x + 2}\right)}{6} \]
   productApply the product rule.✓ Proved
4. \[ = - \frac{\sqrt{2} \left(- 3 e^{3 x + 2} \sin{\left(3 x + \frac{\pi}{4} + 2 \right)} + 3 e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}\right)}{6} \]
   chain algebraApply the chain rule to both terms. Simplify the terms inside the parentheses.✓ Proved
5. \[ = - \frac{\sqrt{2} \left(- e^{3 x + 2} \sin{\left(3 x + \frac{\pi}{4} + 2 \right)} + e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}\right)}{2} \]
   constant-multiple simplifyFactor out the common constant 3. Simplify the fraction 3/6 to 1/2.✓ Proved
6. \[ = - \frac{\sqrt{2} \left(- \sin{\left(3 x + \frac{\pi}{4} + 2 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}\right) e^{3 x + 2}}{2} \]
   algebraFactor out the exponential term.✓ Proved