Derivative of \( \displaystyle \left(2 x - 4\right) e^{x - 1} \)

Problem 2.1014 · medium

1. \[ \frac{d}{d x} \left(2 x - 4\right) e^{x - 1} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \left(2 x - 4\right) \frac{d}{d x} e^{x - 1} + 2 e^{x - 1} \frac{d}{d x} \left(x - 2\right) \]
   productApply the product rule.✓ Proved
3. \[ = \left(2 x - 4\right) \frac{d}{d x} e^{x - 1} + 2 e^{x - 1} \]
   derivative constant-multipleDifferentiate the first part of the product. Simplify the first term.✓ Proved
4. \[ = \left(2 x - 4\right) e^{x - 1} \frac{d}{d x} \left(x - 1\right) + 2 e^{x - 1} \]
   chainApply the chain rule to the exponential term.✓ Proved
5. \[ = \left(2 x - 4\right) e^{x - 1} + 2 e^{x - 1} \]
   derivative constant-multipleDifferentiate the exponent. Simplify the last term.✓ Proved
6. \[ = 2 \left(x - 1\right) e^{x - 1} \]
   algebra simplifyFactor out the common term 2*exp(x - 1). Simplify the expression inside the parentheses.✓ Proved