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

Problem 2.539 · medium

1. \[ \frac{d}{d x} \left(3 x - \frac{3}{4}\right) e^{4 x} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(3 x e^{4 x} - \frac{3 e^{4 x}}{4}\right) \]
   algebraDistribute the term.✓ Proved
3. \[ = \frac{d}{d x} 3 x e^{4 x} - \frac{d}{d x} \frac{3 e^{4 x}}{4} \]
   sumApply the sum rule.✓ Proved
4. \[ = 3 \frac{d}{d x} x e^{4 x} - \frac{3 \frac{d}{d x} e^{4 x}}{4} \]
   constant-multipleFactor out the constants.✓ Proved
5. \[ = 3 x \frac{d}{d x} e^{4 x} + 3 e^{4 x} \frac{d}{d x} x - \frac{3 \frac{d}{d x} e^{4 x}}{4} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 3 x \frac{d}{d x} e^{4 x} + 3 e^{4 x} - \frac{3 \frac{d}{d x} e^{4 x}}{4} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = 12 x e^{4 x} \]
   exponential algebra simplifyDifferentiate exp(4*x). Distribute and simplify the constant multiplication. Combine like terms.✓ Proved