Derivative of \( \displaystyle x^{3} + 6 x^{2} + 12 x \)
Problem 2.1667 · medium Beautiful
Differentiate \( \displaystyle f(x) = x^{3} + 6 x^{2} + 12 x \).
- \[ \frac{d}{d x} \left(x^{3} + 6 x^{2} + 12 x\right) \]sumStart with the derivative of the sum.✓ Proved
- \[ = \frac{d}{d x} 12 x + \frac{d}{d x} 6 x^{2} + \frac{d}{d x} x^{3} \]constant-multipleApply the constant multiple rule to each term.✓ Proved
- \[ = 12 \frac{d}{d x} x + 6 \frac{d}{d x} x^{2} + \frac{d}{d x} x^{3} \]powerApply the power rule to each term.✓ Proved
- \[ = 3 x^{2} + 12 x + 12 \]algebra simplifyEvaluate the derivatives and simplify the terms. Combine terms to get the final answer.✓ Proved
Answer \( 3 \left(x + 2\right)^{2} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 |
| 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: fail (error) — Step 2 is labeled 'constant-multiple' but performs the 'sum' rule (splitting the derivative of a sum into a sum of derivatives). Step 3 is labeled 'power' but performs the 'constant-multiple' rule (pulling out constants 6 and 12) in addition to applying the power rule.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 2 is labeled 'constant-multiple' but performs the 'sum' rule (splitting the derivative of a sum into a sum of derivatives). Step 3 is labeled 'power' but performs the 'constant-multiple' rule (pulling out constants 6 and 12) in addition to applying the power rule.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (style) 2026-10-06 — Step 2 is labeled 'constant-multiple' but actually applies the 'sum' rule to split the derivative of the sum into a sum of derivatives. Step 3 is labeled 'power' but actually applies the 'constant-multiple' rule to pull out the coefficients 6 and 12.
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 2026-10-06 with SymPy 1.14.0.