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

Problem 2.187 · hard Beautiful

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