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

Problem 2.1380 · medium

1. \[ \frac{d}{d x} \left(x - \frac{1}{2}\right) e^{4 x - 1} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(x e^{4 x - 1} - \frac{e^{4 x - 1}}{2}\right) \]
   algebraDistribute the term (x - 1/2).✓ Proved
3. \[ = \frac{d}{d x} x e^{4 x - 1} - \frac{d}{d x} \frac{e^{4 x - 1}}{2} \]
   sumApply the sum rule for differentiation.✓ Proved
4. \[ = \frac{d}{d x} x e^{4 x - 1} - \frac{\frac{d}{d x} e^{4 x - 1}}{2} \]
   constantPull out the constant factor 1/2.✓ Proved
5. \[ = x \frac{d}{d x} e^{4 x - 1} + e^{4 x - 1} \frac{d}{d x} x - \frac{\frac{d}{d x} e^{4 x - 1}}{2} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = x \frac{d}{d x} e^{4 x - 1} + e^{4 x - 1} - \frac{\frac{d}{d x} e^{4 x - 1}}{2} \]
   derivative constantDifferentiate x. Simplify 1*exp(4*x - 1).✓ Proved
7. \[ = \left(x - \frac{1}{2}\right) \frac{d}{d x} e^{4 x - 1} + e^{4 x - 1} \]
   algebraFactor out the common derivative term.✓ Proved
8. \[ = \left(x - \frac{1}{2}\right) e^{4 x - 1} \frac{d}{d x} \left(4 x - 1\right) + e^{4 x - 1} \]
   chainApply the chain rule to the exponential term.✓ Proved
9. \[ = 4 \left(x - \frac{1}{2}\right) e^{4 x - 1} + e^{4 x - 1} \]
   derivativeDifferentiate the inner function 4*x - 1.✓ Proved
10. \[ = \left(4 x - 2\right) e^{4 x - 1} + e^{4 x - 1} \]
    algebraDistribute the 4 into the parentheses.✓ Proved
11. \[ = \left(4 x - 1\right) e^{4 x - 1} \]
    algebra simplifyFactor out the common exp(4*x - 1) term. Combine the terms inside the parentheses.✓ Proved