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

Problem 2.1590 · hard

1. \[ \frac{d}{d x} \left(x e^{3 x} - \frac{e^{3 x}}{3}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x e^{3 x} - \frac{d}{d x} \frac{e^{3 x}}{3} \]
   sum constantApply the sum rule. Factor out the constant 1/3.✓ Proved
3. \[ = x \frac{d}{d x} e^{3 x} + e^{3 x} \frac{d}{d x} x - \frac{d}{d x} \frac{e^{3 x}}{3} \]
   productApply the product rule to the first term.✓ Proved
4. \[ = x \frac{d}{d x} e^{3 x} + e^{3 x} - \frac{d}{d x} \frac{e^{3 x}}{3} \]
   derivativeDifferentiate x.✓ Proved
5. \[ = x \frac{d}{d x} e^{3 x} + e^{3 x} - \frac{\frac{d}{d x} e^{3 x}}{3} \]
   constant-multipleFactor out 1/3 from the third term.✓ Proved
6. \[ = \left(x - \frac{1}{3}\right) \frac{d}{d x} e^{3 x} + e^{3 x} \]
   algebraFactor out the common derivative term.✓ Proved
7. \[ = \left(3 x - 1\right) e^{3 x} + e^{3 x} \]
   chain algebraDifferentiate exp(3*x) using the chain rule. Distribute the 3 into the parentheses.✓ Proved
8. \[ = 3 x e^{3 x} \]
   algebra simplifyFactor out exp(3*x). Simplify the expression inside the parentheses.✓ Proved