Derivative of \( \displaystyle x \left(x^{2} - 9 x + 27\right) \)
Problem 2.151 · hard
Differentiate \( \displaystyle f(x) = x \left(x^{2} - 9 x + 27\right) \).
- \[ \frac{d}{d x} x \left(x^{2} - 9 x + 27\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(x^{2} - 9 x + 27\right) + \left(x^{2} - 9 x + 27\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = x^{2} + x \left(\frac{d}{d x} 27 - \frac{d}{d x} 9 x + \frac{d}{d x} x^{2}\right) - 9 x + 27 \]constant-multipleDifferentiate the first term.✓ Proved
- \[ = x^{2} + x \left(2 x - 9\right) - 9 x + 27 \]derivative simplifyDifferentiate the terms in the second part. Simplify the expression.✓ Proved
- \[ = 3 x^{2} - 18 x + 27 \]algebra simplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( 3 \left(x - 3\right)^{2} \)
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: fail (style) — Step 3 incorrectly labels the application of the linearity of differentiation as "constant-multiple"; it should be "sum". The step also applies a single rule, so the label is the only issue.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 3 applies multiple rules simultaneously: it evaluates the derivative of x, distributes the derivative operator over the sum inside the second term, and applies the constant-multiple rule to the term 9x. This violates the constraint that each step must change only one thing.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies multiple rules simultaneously: it evaluates the derivative of x, distributes the derivative operator over the sum inside the second term, and applies the constant-multiple rule to the term 9x. This violates the constraint that each step must change only one thing.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the linearity of differentiation as "constant-multiple"; it should be "sum". The step also applies a single rule, so the label is the only issue.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies multiple rules at once: it evaluates the derivative of x, distributes the derivative operator over the sum inside the second term (sum rule), and prepares to handle constants, but labels it only as 'constant-multiple'. A single step should only apply one rule.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the derivative to each term inside the parentheses as "constant-multiple"; the correct rule is "sum" (linearity of differentiation).qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 applies multiple rules simultaneously: it differentiates the first term of the product (derivative rule), distributes the derivative operator over the sum in the second term (sum rule), and removes the constant factor 9 (constant-multiple rule). This violates the constraint that each step must change only one thing.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 applies multiple rules simultaneously: it differentiates the first term of the product, distributes the derivative operator over the sum in the second term, and handles the constant multiple inside the sum. This violates the constraint that each step must change only one thing.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 3 applies more than one rule at once (the linearity of differentiation over a sum) and labels it as "constant-multiple" instead of the appropriate "sum" rule. This violates the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 3 applies multiple rules simultaneously: it differentiates the first term of the product (derivative rule), distributes the derivative operator over the sum inside the second term (sum rule), and implicitly handles the constant multiple for 9x. This violates the constraint that each step changes only one thing.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: inconclusive 2026-09-18 — reviewer response could not be parsed: {"verdict":"fail","severity":"style","notes":"Step 3 incorrectly labels the application of the sum rule as \"constant-multiple\". The derivative of the sum \(x^2-9x+27\) should be handled with the \"sgpt-oss:20b: fail 2026-09-17 — Step 3 incorrectly labels the operation as "constant‑multiple" while it actually applies the linearity of differentiation, and step 5’s note claims a simplification when the step is actually a distribution of the factor x. These mismatches could mislead a student.deepseek-r1:70b: pass 2026-09-17
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.