Derivative of \( \displaystyle 3 x \left(4 x^{2} - 18 x + 27\right) \)
Problem 2.472 · hard Video
- \[ \frac{d}{d x} 3 x \left(4 x^{2} - 18 x + 27\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(12 x^{3} - 54 x^{2} + 81 x\right) \]algebraExpand the expression first.✓ Proved
- \[ = \frac{d}{d x} 81 x - \frac{d}{d x} 54 x^{2} + \frac{d}{d x} 12 x^{3} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = 36 x^{2} + \frac{d}{d x} 81 x - \frac{d}{d x} 54 x^{2} \]powerDifferentiate the first term.✓ Proved
- \[ = 36 x^{2} - 108 x + \frac{d}{d x} 81 x \]powerDifferentiate the second term.✓ Proved
- \[ = 36 x^{2} - 108 x + 81 \]power simplifyDifferentiate the third term. Factor the final expression.✓ Proved
✓ 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 |
| 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 expands the polynomial, applies the sum and power rules term-by-term, and simplifies the result. Each step adheres to the one-rule-per-step constraint and uses valid labels.
Senior review claude-sonnet-5-5, 2026-10-09: pass — Every line was proved equal by SymPy, including the final line against the stated answer. The rule labels are legitimate: algebra for the expansion, sum, power for each term, and simplify for factoring out 9. Each step changes one thing. The panel listed no specific objection to rule on.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-09 — Every line was proved equal by SymPy, including the final line against the stated answer. The rule labels are legitimate: algebra for the expansion, sum, power for each term, and simplify for factoring out 9. Each step changes one thing. The panel listed no specific objection to rule on.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly expands the polynomial, applies the sum and power rules term-by-term, and simplifies the result. Each step adheres to the one-rule-per-step constraint and uses valid labels.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly expands the polynomial, applies the sum rule, and differentiates term-by-term using the power rule. The final simplification is algebraically correct.gpt-oss:20b: fail (error) 2026-09-20 — Step 1 incorrectly applies the derivative rule directly to the product 3*x*(4*x**2-18*x+27). It should use the product rule, not a single derivative. This makes the first step mathematically invalid.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly expands the polynomial, applies the sum rule, and differentiates term-by-term using the power rule. The final simplification is algebraically correct.gpt-oss:20b: pass 2026-09-20
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 2026-09-26 with SymPy 1.14.0.