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

Problem 2.292 · hard

1. \[ \frac{d}{d x} 2 x e^{4 x + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 2 \frac{d}{d x} x e^{4 x + 1} \]
   constant-multiplePull out the constant factor 2.✓ Proved
3. \[ = 2 x \frac{d}{d x} e^{4 x + 1} + 2 e^{4 x + 1} \frac{d}{d x} x \]
   productApply the product rule to the remaining terms.✓ Proved
4. \[ = 2 x \frac{d}{d x} e^{4 x + 1} + 2 e^{4 x + 1} \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = 2 x e^{4 x + 1} \frac{d}{d x} \left(4 x + 1\right) + 2 e^{4 x + 1} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = 8 x e^{4 x + 1} + 2 e^{4 x + 1} \]
   derivative algebraDifferentiate the inner function 4*x + 1. Simplify the expression inside the parentheses.✓ Proved
7. \[ = 2 \left(4 x + 1\right) e^{4 x + 1} \]
   algebraFactor out the common exponential term.✓ Proved
8. \[ = \left(8 x + 2\right) e^{4 x + 1} \]
   simplifyFinal simplified form.✓ Proved