Derivative of \( \displaystyle \left(1 - x\right) e^{4 x - 3} \)

Problem 2.283 · hard

1. \[ \frac{d}{d x} \left(1 - x\right) e^{4 x - 3} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \left(1 - x\right) \frac{d}{d x} e^{4 x - 3} + e^{4 x - 3} \frac{d}{d x} \left(1 - x\right) \]
   productApply the product rule.✓ Proved
3. \[ = \left(1 - x\right) \frac{d}{d x} e^{4 x - 3} + e^{4 x - 3} \frac{d}{d x} 1 - e^{4 x - 3} \frac{d}{d x} x \]
   algebraDistribute the exponential term.✓ Proved
4. \[ = \left(1 - x\right) \frac{d}{d x} e^{4 x - 3} - e^{4 x - 3} \]
   derivative simplifyDifferentiate the terms in the parentheses. Simplify the expression.✓ Proved
5. \[ = \left(1 - x\right) e^{4 x - 3} \frac{d}{d x} \left(4 x - 3\right) - e^{4 x - 3} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = 4 \left(1 - x\right) e^{4 x - 3} - e^{4 x - 3} \]
   derivativeDifferentiate the exponent.✓ Proved
7. \[ = \left(3 - 4 x\right) e^{4 x - 3} \]
   algebra algebra simplifyFactor out the common exponential term. Distribute the 4. Combine like terms.✓ Proved