Derivative of \( \displaystyle 3 x \left(x^{2} + 6 x + 12\right) \)
Problem 2.1203 · hard Beautiful Video
Differentiate \( \displaystyle f(x) = 3 x \left(x^{2} + 6 x + 12\right) \).
- \[ \frac{d}{d x} 3 x \left(x^{2} + 6 x + 12\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} x \left(x^{2} + 6 x + 12\right) \]constant-multiplePull out the constant 3.✓ Proved
- \[ = 3 x \frac{d}{d x} \left(x^{2} + 6 x + 12\right) + 3 \left(x^{2} + 6 x + 12\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 3 x^{2} + 3 x \left(\frac{d}{d x} 12 + \frac{d}{d x} 6 x + \frac{d}{d x} x^{2}\right) + 18 x + 36 \]derivativeDifferentiate each term in the sum.✓ Proved
- \[ = 3 x^{2} + 3 x \left(2 x + 6\right) + 18 x + 36 \]derivative algebraEvaluate the derivatives of the power terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = 9 x^{2} + 36 x + 36 \]algebra algebra simplifyDistribute x into the parentheses. Combine like terms. Distribute the 3 to get the final answer.✓ Proved
Answer \( 9 \left(x + 2\right)^{2} \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: pass 2026-09-29
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.