Derivative of \( \displaystyle \left(x + \frac{1}{3}\right) e^{3 x + 2} \)

Problem 2.660 · medium

1. \[ \frac{d}{d x} \left(x + \frac{1}{3}\right) e^{3 x + 2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(x e^{3 x + 2} + \frac{e^{3 x + 2}}{3}\right) \]
   sumDistribute the first term.✓ Proved
3. \[ = \frac{d}{d x} x e^{3 x + 2} + \frac{d}{d x} \frac{e^{3 x + 2}}{3} \]
   sumApply the sum rule.✓ Proved
4. \[ = x \frac{d}{d x} e^{3 x + 2} + e^{3 x + 2} \frac{d}{d x} x + \frac{d}{d x} \frac{e^{3 x + 2}}{3} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = x \frac{d}{d x} e^{3 x + 2} + e^{3 x + 2} + \frac{d}{d x} \frac{e^{3 x + 2}}{3} \]
   derivativeDifferentiate x.✓ Proved
6. \[ = x \frac{d}{d x} e^{3 x + 2} + e^{3 x + 2} + \frac{\frac{d}{d x} e^{3 x + 2}}{3} \]
   constant-multipleFactor out the constant 1/3.✓ Proved
7. \[ = 3 x e^{3 x + 2} + 2 e^{3 x + 2} \]
   chain algebra algebraApply the chain rule to the exponential terms. Simplify the constant multiplication. Combine like terms.✓ Proved
8. \[ = \left(3 x + 2\right) e^{3 x + 2} \]
   simplifyFactor out the common exponential term.✓ Proved