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

Problem 2.653 · hard

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \]
   constant-multiplePull out the constant coefficient.✓ Proved
3. \[ = - \frac{\sqrt{2} \left(e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x - 1}\right)}{2} \]
   productApply the product rule.✓ Proved
4. \[ = - \frac{\sqrt{2} \left(e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]
   derivativeDifferentiate the exponential term.✓ Proved
5. \[ = - \frac{\sqrt{2} \left(- e^{x - 1} \sin{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} \left(x - 1 + \frac{\pi}{4}\right) + e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]
   trigDifferentiate the cosine term.✓ Proved
6. \[ = - \frac{\sqrt{2} \left(- e^{x - 1} \sin{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]
   derivative algebraDifferentiate the inner function of the chain rule. Distribute the negative sign.✓ Proved
7. \[ = - \frac{\sqrt{2} \left(- \sin{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]
   algebraFactor out the exponential term.✓ Proved
8. \[ = e^{x - 1} \sin{\left(x - 1 \right)} \]
   algebra algebra simplify simplifyExpand the trigonometric terms using angle addition formulas. Substitute the value of cos(pi/4) and sin(pi/4). Combine like terms inside the parentheses. Simplify the final expression.✓ Proved