Derivative of \( \displaystyle 5 x e^{4 x} - \frac{5 e^{4 x}}{4} \)

Problem 2.1574 · hard

1. \[ \frac{d}{d x} \left(5 x e^{4 x} - \frac{5 e^{4 x}}{4}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} 5 x e^{4 x} - \frac{d}{d x} \frac{5 e^{4 x}}{4} \]
   sum constant-multipleApply the sum rule. Factor out the constant from the second term.✓ Proved
3. \[ = 5 \frac{d}{d x} x e^{4 x} - \frac{5 \frac{d}{d x} e^{4 x}}{4} \]
   constant-multipleFactor out the constant from the first term.✓ Proved
4. \[ = 5 x \frac{d}{d x} e^{4 x} + 5 e^{4 x} \frac{d}{d x} x - \frac{5 \frac{d}{d x} e^{4 x}}{4} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = 5 x \frac{d}{d x} e^{4 x} + 5 e^{4 x} - \frac{5 \frac{d}{d x} e^{4 x}}{4} \]
   derivativeDifferentiate x.✓ Proved
6. \[ = 20 x e^{4 x} \]
   chain algebra simplifyDifferentiate exp(4*x) using the chain rule. Distribute and simplify the constants. Combine like terms.✓ Proved