Derivative of \( \displaystyle \left(5 x - \frac{20}{3}\right) e^{3 x - 3} \)

Problem 2.213 · hard

1. \[ \frac{d}{d x} \left(5 x - \frac{20}{3}\right) e^{3 x - 3} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(5 x e^{3 x - 3} - \frac{20 e^{3 x - 3}}{3}\right) \]
   algebraDistribute the exponential term.✓ Proved
3. \[ = \frac{d}{d x} 5 x e^{3 x - 3} - \frac{d}{d x} \frac{20 e^{3 x - 3}}{3} \]
   sumApply the linearity of the derivative.✓ Proved
4. \[ = 5 \frac{d}{d x} x e^{3 x - 3} - \frac{20 \frac{d}{d x} e^{3 x - 3}}{3} \]
   constant-multipleFactor out the constants.✓ Proved
5. \[ = 5 x \frac{d}{d x} e^{3 x - 3} + 5 e^{3 x - 3} \frac{d}{d x} x - \frac{20 \frac{d}{d x} e^{3 x - 3}}{3} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 5 x \frac{d}{d x} e^{3 x - 3} + 5 e^{3 x - 3} - \frac{20 \frac{d}{d x} e^{3 x - 3}}{3} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = 15 x e^{3 x - 3} - 15 e^{3 x - 3} \]
   chain algebraApply the chain rule to the exponential term. Simplify the expression by distributing and multiplying.✓ Proved
8. \[ = \left(15 x - 15\right) e^{3 x - 3} \]
   simplifyCombine like terms.✓ Proved