Derivative of \( \displaystyle - x e^{3 x} + \frac{e^{3 x}}{3} \)

Problem 2.1781 · hard

1. \[ \frac{d}{d x} \left(- x e^{3 x} + \frac{e^{3 x}}{3}\right) \]
   sumStart with the derivative of the entire expression.✓ Proved
2. \[ = \frac{d}{d x} \left(- x e^{3 x}\right) + \frac{d}{d x} \frac{e^{3 x}}{3} \]
   sum constant-multipleSplit the derivative into two parts using the sum rule. Pull out the constant 1/3 from the second term.✓ Proved
3. \[ = \frac{d}{d x} \left(- x e^{3 x}\right) + \frac{\frac{d}{d x} e^{3 x}}{3} \]
   constant-multiplePull out the constant -1 from the first term.✓ Proved
4. \[ = - x \frac{d}{d x} e^{3 x} - e^{3 x} \frac{d}{d x} x + \frac{\frac{d}{d x} e^{3 x}}{3} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = - x \frac{d}{d x} e^{3 x} - e^{3 x} + \frac{\frac{d}{d x} e^{3 x}}{3} \]
   derivativeDifferentiate x.✓ Proved
6. \[ = - 3 x e^{3 x} \]
   chain algebra simplifyApply the chain rule to exp(3*x). Simplify the products and constants. Combine like terms.✓ Proved