Derivative of \( \displaystyle 5 x e^{4 x + 1} \)

Problem 2.94 · hard

1. \[ \frac{d}{d x} 5 x e^{4 x + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 5 \frac{d}{d x} x e^{4 x + 1} \]
   constant-multiplePull out the constant factor 5.✓ Proved
3. \[ = 5 x \frac{d}{d x} e^{4 x + 1} + 5 e^{4 x + 1} \frac{d}{d x} x \]
   productApply the product rule to the remaining term.✓ Proved
4. \[ = 5 x \frac{d}{d x} e^{4 x + 1} + 5 e^{4 x + 1} \]
   derivativeDifferentiate x.✓ Proved
5. \[ = 5 x e^{4 x + 1} \frac{d}{d x} \left(4 x + 1\right) + 5 e^{4 x + 1} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = 20 x e^{4 x + 1} + 5 e^{4 x + 1} \]
   derivative algebraDifferentiate the inner function 4*x + 1. Rearrange the terms.✓ Proved
7. \[ = 5 \left(4 x + 1\right) e^{4 x + 1} \]
   simplifyFactor out the common exponential term.✓ Proved