Derivative of \( \displaystyle - \frac{\sqrt{2} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10} \)

Problem 2.609 · hard

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{\sqrt{2} \left(e^{5 x - 1} \frac{d}{d x} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} + \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{5 x - 1}\right)}{10} \]
   productApply the product rule.✓ Proved
4. \[ = - \frac{\sqrt{2} \left(e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} \left(5 x - 1\right) + e^{5 x - 1} \cos{\left(- 5 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} \left(- 5 x + \frac{\pi}{4} + 1\right)\right)}{10} \]
   chainApply the chain rule to both terms.✓ Proved
5. \[ = - \frac{\sqrt{2} \left(5 e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} - 5 e^{5 x - 1} \cos{\left(- 5 x + \frac{\pi}{4} + 1 \right)}\right)}{10} \]
   derivative algebraEvaluate the internal derivatives. Simplify the expression inside the parentheses.✓ Proved
6. \[ = - \frac{\sqrt{2} \left(e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} - e^{5 x - 1} \cos{\left(- 5 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]
   algebra simplifyFactor out the common constant 5. Simplify the fraction 5/10 to 1/2.✓ Proved
7. \[ = - \frac{\sqrt{2} \left(\sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} - \cos{\left(- 5 x + \frac{\pi}{4} + 1 \right)}\right) e^{5 x - 1}}{2} \]
   algebraFactor out the exponential term.✓ Proved