Derivative of \( \displaystyle \left(\frac{1}{5} - x\right) e^{5 x} \)

Problem 2.1198 · medium

1. \[ \frac{d}{d x} \left(\frac{1}{5} - x\right) e^{5 x} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x e^{5 x} + \frac{e^{5 x}}{5}\right) \]
   algebraDistribute the exponential term.✓ Proved
3. \[ = - \frac{d}{d x} x e^{5 x} + \frac{d}{d x} \frac{e^{5 x}}{5} \]
   sumApply the difference rule.✓ Proved
4. \[ = - \frac{d}{d x} x e^{5 x} + \frac{\frac{d}{d x} e^{5 x}}{5} \]
   constant-multipleFactor out the constant 1/5.✓ 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 second 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 simplifyDifferentiate the exponential term using the chain rule. Simplify the coefficients. Distribute the negative sign. Combine like terms.✓ Proved