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

Problem 2.15 · hard

1. \[ \frac{d}{d x} \frac{\left(4 x - 1\right) e^{4 x}}{2} \]
   derivative constant-multipleStart with the derivative of the function. Pull out the constant factor 1/2.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(4 x - 1\right) e^{4 x}}{2} \]
   constant-multipleMove the constant to the front.✓ Proved
3. \[ = \frac{\left(4 x - 1\right) \frac{d}{d x} e^{4 x}}{2} + \frac{e^{4 x} \frac{d}{d x} \left(4 x - 1\right)}{2} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{\left(4 x - 1\right) \frac{d}{d x} e^{4 x}}{2} + 2 e^{4 x} \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = 2 \left(4 x - 1\right) e^{4 x} + 2 e^{4 x} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = \frac{\left(16 x - 4\right) e^{4 x}}{2} + 2 e^{4 x} \]
   algebraDistribute the 4 into the parentheses.✓ Proved
7. \[ = 8 x e^{4 x} \]
   algebra simplify simplifyExpand the expression. Combine like terms. Multiply by the constant 1/2.✓ Proved