Derivative of \( \displaystyle x \left(- 4 x^{2} + 18 x - 27\right) \)
Problem 2.885 · hard
Differentiate \( \displaystyle f(x) = x \left(- 4 x^{2} + 18 x - 27\right) \).
- \[ \frac{d}{d x} x \left(- 4 x^{2} + 18 x - 27\right) \]Start with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(- 4 x^{2} + 18 x - 27\right) + \left(- 4 x^{2} + 18 x - 27\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = - 4 x^{2} + x \frac{d}{d x} \left(- 4 x^{2} + 18 x - 27\right) + 18 x - 27 \]derivative constant-multipleDifferentiate the first part of the product. Simplify the derivative of x.✓ Proved
- \[ = - 4 x^{2} + x \left(- \frac{d}{d x} 27 + \frac{d}{d x} 18 x + \frac{d}{d x} \left(- 4 x^{2}\right)\right) + 18 x - 27 \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = - 4 x^{2} + x \left(18 - 8 x\right) + 18 x - 27 \]derivative constantDifferentiate each term in the sum. The derivative of the constant 27 is 0.✓ Proved
- \[ = - 12 x^{2} + 36 x - 27 \]algebra simplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( - 12 x^{2} + 36 x - 27 \)
✓ 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.
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: fail (style) — Step 4 is labeled "constant-multiple" but the operation is a simple algebraic simplification of multiplying by 1. The correct label should be "algebra".qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, sum rule, and power rule in distinct steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, sum rule, and power rule in distinct steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (style) 2026-09-26 — Step 4 is labeled "constant-multiple" but the operation is a simple algebraic simplification of multiplying by 1. The correct label should be "algebra".qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, sum rule, and power rule in separate steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-26
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.