Derivative of \( \displaystyle \left(3 x - 3\right) e^{6 x - 6} \)

Problem 2.506 · medium

1. \[ \frac{d}{d x} \left(3 x - 3\right) e^{6 x - 6} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \left(3 x - 3\right) \frac{d}{d x} e^{6 x - 6} + e^{6 x - 6} \frac{d}{d x} \left(3 x - 3\right) \]
   productApply the product rule.✓ Proved
3. \[ = \left(3 x - 3\right) \frac{d}{d x} e^{6 x - 6} + 3 e^{6 x - 6} \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = \left(3 x - 3\right) e^{6 x - 6} \frac{d}{d x} \left(6 x - 6\right) + 3 e^{6 x - 6} \]
   chainApply the chain rule to the exponential term.✓ Proved
5. \[ = 6 \left(3 x - 3\right) e^{6 x - 6} + 3 e^{6 x - 6} \]
   derivativeDifferentiate the inner function 6*x - 6.✓ Proved
6. \[ = \left(18 x - 18\right) e^{6 x - 6} + 3 e^{6 x - 6} \]
   algebraDistribute the 6 into the parentheses.✓ Proved
7. \[ = \left(18 x - 15\right) e^{6 x - 6} \]
   algebra simplifyFactor out the common exponential term. Combine the constant terms.✓ Proved