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

Problem 2.81 · hard

1. \[ \frac{d}{d x} x e^{2 x + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} e^{2 x + 1} + e^{2 x + 1} \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = x \frac{d}{d x} e^{2 x + 1} + e^{2 x + 1} \]
   derivative constantDifferentiate the first part of the product. The derivative of x is 1.✓ Proved
4. \[ = x e^{2 x + 1} \frac{d}{d x} \left(2 x + 1\right) + e^{2 x + 1} \]
   chainApply the chain rule to the exponential term.✓ Proved
5. \[ = 2 x e^{2 x + 1} + e^{2 x + 1} \]
   derivative algebraDifferentiate the inner function 2*x + 1. Rearrange the terms.✓ Proved
6. \[ = \left(2 x + 1\right) e^{2 x + 1} \]
   simplifyFactor out the common exponential term.✓ Proved