Derivative of \( \displaystyle \left(4 x + 2\right) e^{8 x + 4} \)

Problem 2.381 · medium

1. \[ \frac{d}{d x} \left(4 x + 2\right) e^{8 x + 4} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \left(4 x + 2\right) \frac{d}{d x} e^{8 x + 4} + e^{8 x + 4} \frac{d}{d x} \left(4 x + 2\right) \]
   productApply the product rule.✓ Proved
3. \[ = \left(4 x + 2\right) \frac{d}{d x} e^{8 x + 4} + 4 e^{8 x + 4} \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = \left(4 x + 2\right) e^{8 x + 4} \frac{d}{d x} \left(8 x + 4\right) + 4 e^{8 x + 4} \]
   chainApply the chain rule to the exponential term.✓ Proved
5. \[ = 8 \left(4 x + 2\right) e^{8 x + 4} + 4 e^{8 x + 4} \]
   derivativeDifferentiate the inner function 8*x + 4.✓ Proved
6. \[ = \left(32 x + 16\right) e^{8 x + 4} + 4 e^{8 x + 4} \]
   algebraDistribute the 8 into the parentheses.✓ Proved
7. \[ = \left(32 x + 20\right) e^{8 x + 4} \]
   algebra simplifyFactor out the common exponential term. Combine the constant terms.✓ Proved