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

Problem 2.175 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\sqrt{2} \frac{d}{d x} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{\sqrt{2} \left(e^{5 x} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} \right)} + \cos{\left(5 x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{5 x}\right)}{10} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{\sqrt{2} \left(5 e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)} + e^{5 x} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{10} \]
   derivativeDifferentiate the exponential term.✓ Proved
5. \[ = \frac{\sqrt{2} \left(- 5 e^{5 x} \sin{\left(5 x + \frac{\pi}{4} \right)} + 5 e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{10} \]
   chain algebraApply the chain rule to the cosine term. Distribute the negative sign.✓ Proved
6. \[ = \frac{\sqrt{2} \left(- e^{5 x} \sin{\left(5 x + \frac{\pi}{4} \right)} + e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{2} \]
   algebra simplifyFactor out the common factor of 5. Simplify the constant coefficient.✓ Proved
7. \[ = \frac{\sqrt{2} \left(- \sin{\left(5 x + \frac{\pi}{4} \right)} + \cos{\left(5 x + \frac{\pi}{4} \right)}\right) e^{5 x}}{2} \]
   algebra rewrite algebraFactor out the exponential term. This step is unnecessary; we will use a trigonometric identity instead. Keep the expression as is to apply the sine/cosine addition identity.✓ Proved
8. \[ = - e^{5 x} \sin{\left(5 x \right)} \]
   rewrite algebra algebra simplify simplifyExpand the trigonometric terms using addition formulas. Substitute cos(pi/4) = sin(pi/4) = sqrt(2)/2. Distribute the negative sign into the parenthesis. Combine like terms. Final simplification of the coefficients.✓ Proved