Derivative of \( \displaystyle 2 x^{2} - 2 x \)
Problem 2.1570 · medium
Differentiate \( \displaystyle f(x) = 2 x^{2} - 2 x \).
- \[ \frac{d}{d x} \left(2 x^{2} - 2 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x^{2} \]sumApply the sum rule.✓ Proved
- \[ = - 2 \frac{d}{d x} x + 2 \frac{d}{d x} x^{2} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = 2 \frac{d}{d x} x^{2} - 2 \]derivativeDifferentiate the second term.✓ Proved
- \[ = 4 x - 2 \]power algebraApply the power rule to x**2. Multiply the constants.✓ Proved
Answer \( 4 x - 2 \)
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 |
| 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 two rules at once: it uses the power rule to replace Derivative(x**2) with 2*x and then immediately multiplies by the outer constant 2. This violates the rule that each step must change only one thing.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: fail (error) 2026-10-05 — Step 5 applies two rules at once: it uses the power rule to replace Derivative(x**2) with 2*x and then immediately multiplies by the outer constant 2. This violates the rule that each step must change only one thing.qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.