Derivative of \( \displaystyle \left(x - \frac{4}{3}\right) e^{3 x - 3} \)

Problem 2.967 · medium

1. \[ \frac{d}{d x} \left(x - \frac{4}{3}\right) e^{3 x - 3} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(x e^{3 x - 3} - \frac{4 e^{3 x - 3}}{3}\right) \]
   algebraDistribute the term (x - 4/3).✓ Proved
3. \[ = \frac{d}{d x} x e^{3 x - 3} - \frac{d}{d x} \frac{4 e^{3 x - 3}}{3} \]
   sumApply the difference rule.✓ Proved
4. \[ = \frac{d}{d x} x e^{3 x - 3} - \frac{4 \frac{d}{d x} e^{3 x - 3}}{3} \]
   constantPull out the constant 4/3.✓ Proved
5. \[ = - \frac{4 e^{3 x - 3} \frac{d}{d x} \left(3 x - 3\right)}{3} + \frac{d}{d x} x e^{3 x - 3} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = - 4 e^{3 x - 3} + \frac{d}{d x} x e^{3 x - 3} \]
   derivative algebraDifferentiate the inner function 3*x - 3. Simplify the constant multiplication.✓ Proved
7. \[ = x \frac{d}{d x} e^{3 x - 3} + e^{3 x - 3} \frac{d}{d x} x - 4 e^{3 x - 3} \]
   productApply the product rule to the first term.✓ Proved
8. \[ = x \frac{d}{d x} e^{3 x - 3} - 3 e^{3 x - 3} \]
   derivativeDifferentiate x.✓ Proved
9. \[ = x e^{3 x - 3} \frac{d}{d x} \left(3 x - 3\right) - 3 e^{3 x - 3} \]
   chainApply the chain rule to the exponential term.✓ Proved
10. \[ = 3 x e^{3 x - 3} - 3 e^{3 x - 3} \]
    derivative algebraDifferentiate the inner function 3*x - 3. Rearrange the terms.✓ Proved
11. \[ = \left(3 x - 3\right) e^{3 x - 3} \]
    algebra simplifyFactor out the common exponential term. Simplify the expression in the parentheses.✓ Proved