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

Problem 2.1224 · medium

1. \[ \frac{d}{d x} \left(x + \frac{1}{4}\right) e^{4 x + 2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x e^{4 x + 2} + \frac{d}{d x} \frac{e^{4 x + 2}}{4} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} x e^{4 x + 2} + \frac{\frac{d}{d x} e^{4 x + 2}}{4} \]
   constant-multipleApply the constant multiple rule to the second term.✓ Proved
4. \[ = x \frac{d}{d x} e^{4 x + 2} + e^{4 x + 2} \frac{d}{d x} x + \frac{\frac{d}{d x} e^{4 x + 2}}{4} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = x \frac{d}{d x} e^{4 x + 2} + e^{4 x + 2} + \frac{\frac{d}{d x} e^{4 x + 2}}{4} \]
   derivativeDifferentiate x.✓ Proved
6. \[ = 4 x e^{4 x + 2} + 2 e^{4 x + 2} \]
   chain algebra algebraApply the chain rule to the exponential terms. Simplify the coefficients. Combine like terms.✓ Proved
7. \[ = \left(4 x + 2\right) e^{4 x + 2} \]
   algebraFactor out the exponential term.✓ Proved