Derivative of \( \displaystyle 3 x \left(4 x^{2} - 18 x + 27\right) \)
Problem 2.472 · hard
Differentiate \( \displaystyle f(x) = 3 x \left(4 x^{2} - 18 x + 27\right) \).
- \[ \frac{d}{d x} 3 x \left(4 x^{2} - 18 x + 27\right) \]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
Answer \( 9 \left(2 x - 3\right)^{2} \)
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 |
| 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.
Every verdict on record (6)
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.