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

Problem 2.1751 · hard

1. \[ \frac{d}{d x} \left(x e^{4 x} - \frac{e^{4 x}}{4}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x e^{4 x} - \frac{d}{d x} \frac{e^{4 x}}{4} \]
   sum constantApply the sum rule. Factor out the constant 1/4.✓ Proved
3. \[ = x \frac{d}{d x} e^{4 x} + e^{4 x} \frac{d}{d x} x - \frac{d}{d x} \frac{e^{4 x}}{4} \]
   productApply the product rule to the first term.✓ Proved
4. \[ = x \frac{d}{d x} e^{4 x} + e^{4 x} - \frac{d}{d x} \frac{e^{4 x}}{4} \]
   derivative simplifyDifferentiate x. Simplify the first term.✓ Proved
5. \[ = 4 x e^{4 x} + e^{4 x} - \frac{d}{d x} \frac{e^{4 x}}{4} \]
   chain algebraDifferentiate exp(4*x) using the chain rule. Simplify the second term.✓ Proved
6. \[ = 4 x e^{4 x} \]
   chain algebra simplifyDifferentiate the last term using the chain rule. Simplify the last term. Combine like terms.✓ Proved