Derivative of \( \displaystyle x \left(16 x^{2} - 36 x + 27\right) \)
Problem 2.1026 · hard
Differentiate \( \displaystyle f(x) = x \left(16 x^{2} - 36 x + 27\right) \).
- \[ \frac{d}{d x} x \left(16 x^{2} - 36 x + 27\right) \]Start with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(16 x^{2} - 36 x + 27\right) + \left(16 x^{2} - 36 x + 27\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 16 x^{2} + x \left(\frac{d}{d x} 27 - \frac{d}{d x} 36 x + \frac{d}{d x} 16 x^{2}\right) - 36 x + 27 \]constantDifferentiate the first term.✓ Proved
- \[ = 16 x^{2} + x \left(\frac{d}{d x} 16 x^{2} - 36\right) - 36 x + 27 \]constantDifferentiate the terms in the second part.✓ Proved
- \[ = 16 x^{2} + x \left(32 x - 36\right) - 36 x + 27 \]power algebraDifferentiate the power term 16*x**2. Simplify the first part.✓ Proved
- \[ = 48 x^{2} - 72 x + 27 \]algebra simplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( 3 \left(4 x - 3\right)^{2} \)
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 |
| 8 | ✓ 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 3 incorrectly labels the application of the derivative as "constant"; the correct label is "derivative". The step also applies the derivative to each term inside the parentheses, which is acceptable, but the label does not match the rule used.qwen3.6:27b-mlx: fail (error) — Step 3 applies the derivative of x (derivative rule) and distributes the derivative operator over the sum (sum rule) simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'constant' is incorrect for differentiating x.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 3 applies the derivative of x (derivative rule) and distributes the derivative operator over the sum (sum rule) simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'constant' is incorrect for differentiating x.gpt-oss:20b: fail (style) 2026-09-27 — Step 3 incorrectly labels the application of the derivative as "constant"; the correct label is "derivative". The step also applies the derivative to each term inside the parentheses, which is acceptable, but the label does not match the rule used.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (style) 2026-09-27 — Step 3 incorrectly labels the differentiation of the first term (x) as "constant"; it should be "derivative". The note also misstates that the first term is being differentiated, whereas it is the derivative of x that is applied. This labeling error could mislead a student about which rule applies to a simple linear term.
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 2026-09-27 with SymPy 1.14.0.