Derivative of \( \displaystyle x e^{2 x} - \frac{e^{2 x}}{2} \)

Problem 2.1572 · hard

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