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

Problem 2.1749 · hard

1. \[ \frac{d}{d x} \left(- x e^{4 x} + \frac{e^{4 x}}{4}\right) \]
   sumStart with the derivative of the entire function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x e^{4 x}\right) + \frac{d}{d x} \frac{e^{4 x}}{4} \]
   sum constant-multiple constant-multiple productApply the sum rule. Factor out the constant 1/4. Factor out the negative sign. Apply the product rule to the first term.✓ Proved
3. \[ = - x \frac{d}{d x} e^{4 x} - e^{4 x} \frac{d}{d x} x + \frac{d}{d x} \frac{e^{4 x}}{4} \]
   productExpand the product rule.✓ Proved
4. \[ = - x \frac{d}{d x} e^{4 x} - e^{4 x} + \frac{d}{d x} \frac{e^{4 x}}{4} \]
   derivativeDifferentiate x.✓ Proved
5. \[ = - x e^{4 x} \frac{d}{d x} 4 x - e^{4 x} + \frac{d}{d x} \frac{e^{4 x}}{4} \]
   chainApply the chain rule to exp(4*x).✓ Proved
6. \[ = - 4 x e^{4 x} - e^{4 x} + \frac{d}{d x} \frac{e^{4 x}}{4} \]
   derivative algebraDifferentiate 4*x. Simplify the expression inside the parentheses.✓ Proved
7. \[ = - 4 x e^{4 x} - e^{4 x} + \frac{\frac{d}{d x} e^{4 x}}{4} \]
   constant-multipleFactor out the constant 1/4 from the second term.✓ Proved
8. \[ = - 4 x e^{4 x} + e^{4 x} \frac{d}{d x} x - e^{4 x} \]
   chainApply the chain rule to the second term.✓ Proved
9. \[ = - 4 x e^{4 x} \]
   derivative algebra simplifySimplify the derivative of the second term. Distribute the negative sign. Combine like terms.✓ Proved