Derivative of \( \displaystyle 4 x \left(- x^{2} - 3 x - 3\right) \)

Problem 2.1236 · hard

1. \[ \frac{d}{d x} 4 x \left(- x^{2} - 3 x - 3\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 4 \frac{d}{d x} x \left(- x^{2} - 3 x - 3\right) \]
   constant-multiplePull out the constant factor 4.✓ Proved
3. \[ = 4 x \frac{d}{d x} \left(- x^{2} - 3 x - 3\right) + 4 \left(- x^{2} - 3 x - 3\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
4. \[ = - 4 x^{2} + 4 x \frac{d}{d x} \left(- x^{2} - 3 x - 3\right) - 12 x - 12 \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = - 4 x^{2} + 4 x \frac{d}{d x} \left(-3\right) + 4 x \frac{d}{d x} \left(- 3 x\right) + 4 x \frac{d}{d x} \left(- x^{2}\right) - 12 x - 12 \]
   sumDifferentiate the second part of the product using the sum rule.✓ Proved
6. \[ = - 12 x^{2} - 24 x - 12 \]
   derivative algebra algebra simplifyDifferentiate each term in the sum. Simplify the expression inside the parentheses. Combine like terms. Distribute the 4 and finalize the result.✓ Proved