Derivative of \( \displaystyle x \left(25 x^{2} - 45 x + 27\right) \)
Problem 2.1181 · hard
Differentiate \( \displaystyle f(x) = x \left(25 x^{2} - 45 x + 27\right) \).
- \[ \frac{d}{d x} x \left(25 x^{2} - 45 x + 27\right) \]Start with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(25 x^{2} - 45 x + 27\right) + \left(25 x^{2} - 45 x + 27\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 25 x^{2} + x \left(\frac{d}{d x} 27 - \frac{d}{d x} 45 x + \frac{d}{d x} 25 x^{2}\right) - 45 x + 27 \]derivativeDifferentiate each term in the second factor.✓ Proved
- \[ = 25 x^{2} + x \left(\frac{d}{d x} 25 x^{2} - 45\right) - 45 x + 27 \]constant constantDifferentiate the constant terms. Differentiate the linear term.✓ Proved
- \[ = 25 x^{2} + x \left(50 x - 45\right) - 45 x + 27 \]power algebraDifferentiate the quadratic term. Simplify the first term.✓ Proved
- \[ = 75 x^{2} - 90 x + 27 \]algebra simplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( 3 \left(5 x - 3\right)^{2} \)
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 |
| 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: passqwen3.6:27b-mlx: fail (error) — Step 3 applies the sum rule to split the derivative of the polynomial, but labels it 'derivative'. The label 'derivative' is reserved for unfolding d/dx on a known form (like d/dx(x^n)), not for structural decomposition of sums. Additionally, Step 3 performs two distinct operations: differentiating the first factor (x -> 1) and splitting the second factor's derivative, violating the one-rule-per-step constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 applies the sum rule to split the derivative of the polynomial, but labels it 'derivative'. The label 'derivative' is reserved for unfolding d/dx on a known form (like d/dx(x^n)), not for structural decomposition of sums. Additionally, Step 3 performs two distinct operations: differentiating the first factor (x -> 1) and splitting the second factor's derivative, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 applies the sum rule to split the derivative of the polynomial, but labels it 'derivative'. The label 'derivative' is reserved for unfolding d/dx on a known form (like d/dx(x^n)), not for linearity/sum rules. Furthermore, Step 3 performs two distinct operations: differentiating the first factor (x -> 1) and splitting the second factor's derivative. This violates the 'one rule per step' constraint.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.