Derivative of \( \displaystyle 12 x \left(x^{2} + 3 x + 3\right) \)

Problem 2.176 · hard Beautiful

1. \[ \frac{d}{d x} 12 x \left(x^{2} + 3 x + 3\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 12 \frac{d}{d x} x \left(x^{2} + 3 x + 3\right) \]
   constant-multiplePull out the constant factor 12.✓ Proved
3. \[ = 12 x \frac{d}{d x} \left(x^{2} + 3 x + 3\right) + 12 \left(x^{2} + 3 x + 3\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
4. \[ = 12 x^{2} + 12 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 3 x + \frac{d}{d x} x^{2}\right) + 36 x + 36 \]
   sumApply the sum rule to the terms inside the parentheses.✓ Proved
5. \[ = 12 x^{2} + 12 x \left(2 x + 3\right) + 36 x + 36 \]
   derivative algebraDifferentiate each term individually. Simplify the expression inside the parentheses.✓ Proved
6. \[ = 36 x^{2} + 72 x + 36 \]
   algebra algebra simplifyDistribute x into the parentheses. Combine like terms. Distribute the 12 to get the final simplified form.✓ Proved