Derivative of \( \displaystyle - x e^{5 x} + \frac{e^{5 x}}{5} \)

Problem 2.1832 · hard

1. \[ \frac{d}{d x} \left(- x e^{5 x} + \frac{e^{5 x}}{5}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x e^{5 x}\right) + \frac{d}{d x} \frac{e^{5 x}}{5} \]
   sum constantApply the sum rule. Factor out the constant 1/5.✓ Proved
3. \[ = \frac{d}{d x} \left(- x e^{5 x}\right) + \frac{\frac{d}{d x} e^{5 x}}{5} \]
   constant-multiple constant algebraApply the constant multiple rule to the second term. Factor out the negative sign. Group terms for the product rule.✓ Proved
4. \[ = - \frac{d}{d x} x e^{5 x} + \frac{\frac{d}{d x} e^{5 x}}{5} \]
   constant-multipleFactor out the -1 from the first term.✓ Proved
5. \[ = - x \frac{d}{d x} e^{5 x} - e^{5 x} \frac{d}{d x} x + \frac{\frac{d}{d x} e^{5 x}}{5} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = - x \frac{d}{d x} e^{5 x} - e^{5 x} + \frac{\frac{d}{d x} e^{5 x}}{5} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = - 5 x e^{5 x} \]
   chain algebra algebra simplifyApply the chain rule to exp(5*x). Simplify the terms inside and outside the parentheses. Distribute the -1. Combine like terms.✓ Proved