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

Problem 2.1682 · hard

1. \[ \frac{d}{d x} \left(5 x e^{5 x} - e^{5 x}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} 5 x e^{5 x} - \frac{d}{d x} e^{5 x} \]
   sumApply the difference rule.✓ Proved
3. \[ = 5 \frac{d}{d x} x e^{5 x} - \frac{d}{d x} e^{5 x} \]
   constant-multipleFactor out the constant 5 from the first term.✓ Proved
4. \[ = 5 x \frac{d}{d x} e^{5 x} + 5 e^{5 x} \frac{d}{d x} x - \frac{d}{d x} e^{5 x} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = 5 x \frac{d}{d x} e^{5 x} + 5 e^{5 x} - \frac{d}{d x} e^{5 x} \]
   derivativeDifferentiate x.✓ Proved
6. \[ = 25 x e^{5 x} \]
   chain product simplifyApply the chain rule to exp(5*x). Distribute the 5 into the parentheses. Combine like terms.✓ Proved