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

Problem 2.883 · hard

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}}{2}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{\sqrt{2} \left(e^{x + 2} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 2 \right)} + \cos{\left(x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} e^{x + 2}\right)}{2} \]
   productApply the product rule.✓ Proved
4. \[ = - \frac{\sqrt{2} \left(e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)} + e^{x + 2} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 2 \right)}\right)}{2} \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = - \frac{\sqrt{2} \left(- e^{x + 2} \sin{\left(x + \frac{\pi}{4} + 2 \right)} + e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}\right)}{2} \]
   derivativeDifferentiate the second part of the product.✓ Proved
6. \[ = - \frac{\sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} + 2 \right)} + \cos{\left(x + \frac{\pi}{4} + 2 \right)}\right) e^{x + 2}}{2} \]
   algebra rewrite simplifyFactor out the common term exp(x + 2). Use trigonometric identities if needed, but here we simplify the expression structure. The expression is already in its simplest form.✓ Proved