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

Problem 2.89 · hard

1. \[ \frac{d}{d x} \left(- x e^{2 x + 1}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - x \frac{d}{d x} e^{2 x + 1} + e^{2 x + 1} \frac{d}{d x} \left(- x\right) \]
   productApply the product rule.✓ Proved
3. \[ = - x \frac{d}{d x} e^{2 x + 1} - e^{2 x + 1} \]
   derivativeDifferentiate the first part of the product.✓ 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. Simplify the expression.✓ Proved
6. \[ = - \left(2 x + 1\right) e^{2 x + 1} \]
   simplifyFactor out the common term.✓ Proved