Derivative of \( \displaystyle - \frac{3 \sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \)

Problem 2.342 · hard

1. \[ \frac{d}{d x} \left(- \frac{3 \sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{3 \sqrt{2} \frac{d}{d x} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{3 \sqrt{2} \left(e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x - 1}\right)}{2} \]
   productApply the product rule.✓ Proved
4. \[ = - \frac{3 \sqrt{2} \left(e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]
   derivativeDifferentiate exp(x - 1).✓ Proved
5. \[ = - \frac{3 \sqrt{2} \left(- e^{x - 1} \sin{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]
   trigDifferentiate cos(x - 1 + pi/4).✓ Proved
6. \[ = - \frac{3 \sqrt{2} \left(- \sin{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]
   algebraFactor out exp(x - 1).✓ Proved
7. \[ = - \frac{3 \sqrt{2} \left(- \cos{\left(- x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]
   algebraUse the identity sin(theta) = cos(pi/2 - theta) or similar, but let's simplify the expression directly.✓ Proved
8. \[ = - \frac{3 \sqrt{2} \left(- \sin{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]
   simplifyThe expression is already in a standard form, but let's combine terms if possible. Actually, let's use the identity cos(a) - sin(a) = sqrt(2)cos(a + pi/4).✓ Proved
9. \[ = 3 e^{x - 1} \sin{\left(x - 1 \right)} \]
   algebra algebra simplify trig simplifyUse the identity cos(u) - sin(u) = sqrt(2)cos(u + pi/4). Simplify the argument inside the cosine. Multiply the constants: (sqrt(2)*sqrt(2))/2 = 1. Use the identity cos(u + pi/2) = -sin(u). Simplify the final expression.✓ Proved