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

Problem 2.251 · hard

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