Derivative of \( \displaystyle 3 x e^{3 x + 1} \)

Problem 2.390 · medium

1. \[ \frac{d}{d x} 3 x e^{3 x + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 3 \frac{d}{d x} x e^{3 x + 1} \]
   constant-multiplePull out the constant factor 3.✓ Proved
3. \[ = 3 x \frac{d}{d x} e^{3 x + 1} + 3 e^{3 x + 1} \frac{d}{d x} x \]
   productApply the product rule to the remaining terms.✓ Proved
4. \[ = 3 x \frac{d}{d x} e^{3 x + 1} + 3 e^{3 x + 1} \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = 3 x e^{3 x + 1} \frac{d}{d x} \left(3 x + 1\right) + 3 e^{3 x + 1} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = 9 x e^{3 x + 1} + 3 e^{3 x + 1} \]
   derivative algebra algebraDifferentiate the inner function 3*x + 1. Simplify the expression inside the parentheses. Distribute the 3.✓ Proved
7. \[ = 3 \left(3 x + 1\right) e^{3 x + 1} \]
   simplifyFactor out the common term.✓ Proved