Derivative of \( \displaystyle - 5 x^{2} + 2 x \)
Problem 2.2078 · medium
Differentiate \( \displaystyle f(x) = - 5 x^{2} + 2 x \).
- \[ \frac{d}{d x} \left(- 5 x^{2} + 2 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 2 x + \frac{d}{d x} \left(- 5 x^{2}\right) \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 2 x - 5 \frac{d}{d x} x^{2} \]constant-multipleApply the constant multiple rule to the first term.✓ Proved
- \[ = 2 - 10 x \]power algebraDifferentiate the power terms. Simplify the coefficients.✓ Proved
Answer \( 2 - 10 x \)
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 |
| 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 (error) — Step 4 applies two rules at once (the power rule for the first term and the constant‑multiple rule for the second term) and is labeled only as "power", which does not reflect the operations performed.qwen3.6:27b-mlx: pass — The solution correctly applies the sum rule, constant multiple rule, and power rule in separate steps. The labels are accurate and the algebraic simplification is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-11 — Step 4 applies two rules at once (the power rule for the first term and the constant‑multiple rule for the second term) and is labeled only as "power", which does not reflect the operations performed.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the sum rule, constant multiple rule, and power rule in separate steps. The labels are accurate and the algebraic simplification is correct.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.