Derivative of \( \displaystyle 6 x \left(3 x^{2} + 6 x + 4\right) \)
Problem 2.1293 · hard
Differentiate \( \displaystyle f(x) = 6 x \left(3 x^{2} + 6 x + 4\right) \).
- \[ \frac{d}{d x} 6 x \left(3 x^{2} + 6 x + 4\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(18 x^{3} + 36 x^{2} + 24 x\right) \]algebraDistribute 6*x into the parentheses.✓ Proved
- \[ = \frac{d}{d x} 24 x + \frac{d}{d x} 36 x^{2} + \frac{d}{d x} 18 x^{3} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = 2 \frac{d}{d x} 12 x + 36 \frac{d}{d x} x^{2} + 18 \frac{d}{d x} x^{3} \]constant-multipleFactor out the constants from each term.✓ Proved
- \[ = 54 x^{2} + 72 x + 24 \]power simplifyApply the power rule to each term. Multiply the constants to get the final result.✓ Proved
Answer \( 6 \left(3 x + 2\right)^{2} \)
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. A reviewer's objection to the explanation stands; see 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 |
| 6 | ✓ 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: fail (error) — Step 4 applies two rules at once: it factors 24 as 2·12 and then pulls out the constant 2, which violates the one‑rule‑per‑step rule. The correct step would be 24*Derivative(x, x) or 2*12*Derivative(x, x) but not both in the same line.qwen3.6:27b-mlx: fail (error) — Step 4 incorrectly applies the constant-multiple rule by changing the term 24*x to 2*12*x and factoring out 2 instead of 24, which is an algebraic error not permitted under the 'constant-multiple' label for derivatives.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 incorrectly applies the constant-multiple rule by changing the term 24*x to 2*12*x and factoring out 2 instead of 24, which is an algebraic error not permitted under the 'constant-multiple' label for derivatives.gpt-oss:20b: fail (error) 2026-10-03 — Step 4 applies two rules at once: it factors 24 as 2·12 and then pulls out the constant 2, which violates the one‑rule‑per‑step rule. The correct step would be 24*Derivative(x, x) or 2*12*Derivative(x, x) but not both in the same line.qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 4 is mathematically incorrect: it transforms 24*x into 2*12*x (changing the coefficient from 24 to 2) while applying the constant-multiple rule, which is not a valid algebraic equivalence.gpt-oss:20b: fail (error) 2026-09-30 — Step 4 applies two transformations at once: it pulls out a constant factor and rewrites the term as 2*Derivative(12*x). Only a single rule may be applied per step, so this step violates the contract.
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.