Derivative of \( \displaystyle 3 x \left(x^{2} + 6 x + 12\right) \)

Problem 2.1203 · hard Beautiful

1. \[ \frac{d}{d x} 3 x \left(x^{2} + 6 x + 12\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 3 \frac{d}{d x} x \left(x^{2} + 6 x + 12\right) \]
   constant-multiplePull out the constant 3.✓ Proved
3. \[ = 3 x \frac{d}{d x} \left(x^{2} + 6 x + 12\right) + 3 \left(x^{2} + 6 x + 12\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
4. \[ = 3 x^{2} + 3 x \left(\frac{d}{d x} 12 + \frac{d}{d x} 6 x + \frac{d}{d x} x^{2}\right) + 18 x + 36 \]
   derivativeDifferentiate each term in the sum.✓ Proved
5. \[ = 3 x^{2} + 3 x \left(2 x + 6\right) + 18 x + 36 \]
   derivative algebraEvaluate the derivatives of the power terms. Simplify the expression inside the parentheses.✓ Proved
6. \[ = 9 x^{2} + 36 x + 36 \]
   algebra algebra simplifyDistribute x into the parentheses. Combine like terms. Distribute the 3 to get the final answer.✓ Proved