Derivative of \( \displaystyle - 3 x^{2} + 2 x \)
Problem 2.2005 · medium
Differentiate \( \displaystyle f(x) = - 3 x^{2} + 2 x \).
- \[ \frac{d}{d x} \left(- 3 x^{2} + 2 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 2 x + \frac{d}{d x} \left(- 3 x^{2}\right) \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 2 x - 3 \frac{d}{d x} x^{2} \]constant-multipleApply the constant multiple rule to the first term.✓ Proved
- \[ = 2 \frac{d}{d x} x - 3 \frac{d}{d x} x^{2} \]constant-multipleApply the constant multiple rule to the second term.✓ Proved
- \[ = 2 - 6 x \]power algebraDifferentiate the power terms. Simplify the expression.✓ Proved
Answer \( 2 - 6 x \)
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 |
| 6 | ✓ 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 5 applies the power rule to two separate terms in one line, violating the rule that each step must change only one thing. It should be split into two separate steps, one for –3*(2*x) and one for 2*(1).qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — Step 5 applies the power rule to two separate terms in one line, violating the rule that each step must change only one thing. It should be split into two separate steps, one for –3*(2*x) and one for 2*(1).qwen3.6:27b-mlx: pass 2026-10-10gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.