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

Problem 2.1693 · 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. \[ = \frac{d}{d x} x e^{5 x} - \frac{d}{d x} \frac{e^{5 x}}{5} \]
   sum algebraApply the sum rule. Factor out the constant 1/5.✓ Proved
3. \[ = x \frac{d}{d x} e^{5 x} + e^{5 x} \frac{d}{d x} x - \frac{d}{d x} \frac{e^{5 x}}{5} \]
   productApply the product rule to the first term.✓ Proved
4. \[ = x \frac{d}{d x} e^{5 x} + e^{5 x} - \frac{d}{d x} \frac{e^{5 x}}{5} \]
   derivativeDifferentiate x.✓ Proved
5. \[ = x \frac{d}{d x} e^{5 x} + e^{5 x} - \frac{\frac{d}{d x} e^{5 x}}{5} \]
   constant-multipleFactor out 1/5 from the third term.✓ Proved
6. \[ = 5 x e^{5 x} \]
   chain algebra simplifyDifferentiate exp(5*x) using the chain rule. Simplify the products. Combine like terms.✓ Proved