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

Problem 2.960 · medium

1. \[ \frac{d}{d x} \frac{\left(4 x + 1\right) e^{4 x + 2}}{2} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(4 x + 1\right) e^{4 x + 2}}{2} \]
   constant-multiplePull out the constant factor 1/2.✓ Proved
3. \[ = \frac{\left(4 x + 1\right) \frac{d}{d x} e^{4 x + 2}}{2} + \frac{e^{4 x + 2} \frac{d}{d x} \left(4 x + 1\right)}{2} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{\left(4 x + 1\right) \frac{d}{d x} e^{4 x + 2}}{2} + 2 e^{4 x + 2} \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = \frac{\left(4 x + 1\right) e^{4 x + 2} \frac{d}{d x} \left(4 x + 2\right)}{2} + 2 e^{4 x + 2} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = 2 \left(4 x + 1\right) e^{4 x + 2} + 2 e^{4 x + 2} \]
   derivativeDifferentiate the exponent 4*x + 2.✓ Proved
7. \[ = \frac{\left(16 x + 4\right) e^{4 x + 2}}{2} + 2 e^{4 x + 2} \]
   algebraDistribute the 4 into the binomial.✓ Proved
8. \[ = \left(8 x + 4\right) e^{4 x + 2} \]
   algebraFactor out the common term exp(4*x + 2).✓ Proved
9. \[ = \frac{\left(16 x + 8\right) e^{4 x + 2}}{2} \]
   algebraCombine the terms inside the parentheses.✓ Proved
10. \[ = \left(8 x + 4\right) e^{4 x + 2} \]
    simplifyDivide the polynomial by 2 to get the final simplified form.✓ Proved