Derivative of \( \displaystyle - \frac{\sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \)

Problem 2.4 · hard

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \]
   constant-multipleFactor out the constant term.✓ Proved
3. \[ = - \frac{\sqrt{2} \left(e^{x + 1} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{x + 1}\right)}{2} \]
   productApply the product rule.✓ Proved
4. \[ = - \frac{\sqrt{2} \left(e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)} + e^{x + 1} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]
   derivativeCompute the derivative of exp(x + 1).✓ Proved
5. \[ = - \frac{\sqrt{2} \left(- e^{x + 1} \sin{\left(x + \frac{\pi}{4} + 1 \right)} + e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]
   derivative algebraCompute the derivative of cos(x + pi/4 + 1). Simplify the expression inside the parentheses.✓ Proved
6. \[ = - \frac{\sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right) e^{x + 1}}{2} \]
   algebraFactor out exp(x + 1).✓ Proved
7. \[ = e^{x + 1} \sin{\left(x + 1 \right)} \]
   simplify algebra algebraUse the trigonometric identity cos(theta) - sin(theta) = sqrt(2)cos(theta + pi/4). Combine the pi/4 terms. Multiply the constants (sqrt(2)/2 * sqrt(2) = 1).✓ Proved