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

Problem 2.1141 · medium

1. \[ \frac{d}{d x} \left(3 x - 3\right) e^{2 x - 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 3 \frac{d}{d x} \left(x - 1\right) e^{2 x - 1} \]
   constant-multiplePull out the constant factor 3.✓ Proved
3. \[ = 3 \left(x - 1\right) \frac{d}{d x} e^{2 x - 1} + 3 e^{2 x - 1} \frac{d}{d x} \left(x - 1\right) \]
   productApply the product rule.✓ Proved
4. \[ = 3 \left(x - 1\right) \frac{d}{d x} e^{2 x - 1} + 3 \left(- \frac{d}{d x} 1 + \frac{d}{d x} x\right) e^{2 x - 1} \]
   sumDifferentiate the terms inside the parenthesis separately.✓ Proved
5. \[ = 3 \left(x - 1\right) \frac{d}{d x} e^{2 x - 1} + 3 e^{2 x - 1} \]
   derivative simplifyEvaluate the derivatives of x and 1. Simplify the expression inside the parentheses.✓ Proved
6. \[ = 3 \left(x - 1\right) e^{2 x - 1} \frac{d}{d x} \left(2 x - 1\right) + 3 e^{2 x - 1} \]
   chainApply the chain rule to the exponential term.✓ Proved
7. \[ = 6 \left(x - 1\right) e^{2 x - 1} + 3 e^{2 x - 1} \]
   derivativeEvaluate the derivative of the exponent.✓ Proved
8. \[ = 3 \left(2 x - 2\right) e^{2 x - 1} + 3 e^{2 x - 1} \]
   algebraRearrange the terms.✓ Proved
9. \[ = 3 \left(2 x - 1\right) e^{2 x - 1} \]
   algebra algebra simplifyFactor out the common exponential term. Distribute the 2 into the parenthesis. Combine like terms to get the final answer.✓ Proved