Derivative of \( \displaystyle 2 x \left(x - 1\right) \)
Problem 2.1304 · medium
Differentiate \( \displaystyle f(x) = 2 x \left(x - 1\right) \).
- \[ \frac{d}{d x} 2 x \left(x - 1\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(2 x^{2} - 2 x\right) \]algebraExpand the expression.✓ Proved
- \[ = - 2 \frac{d}{d x} x + 2 \frac{d}{d x} x^{2} \]sumApply the sum rule.✓ Proved
- \[ = 2 \frac{d}{d x} x^{2} - 2 \]constantThe derivative of a constant is zero.✓ Proved
- \[ = 4 x - 2 \]power simplifyApply the power rule to x**2. Simplify the final expression.✓ 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. A reviewer's objection to the explanation stands; see 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 1 applies the derivative rule directly to the product 2*x*(x‑1) instead of using the product rule, so the first step is incorrect. Step 4 then incorrectly treats the derivative of x as a constant, yielding 0; the correct derivative is 1. These errors invalidate the solution.qwen3.6:27b-mlx: fail (error) — Step 3 applies both the sum rule and the constant-multiple rule simultaneously, violating the one-rule-per-step constraint. Step 4 incorrectly labels the differentiation of x as 'constant'; it should be 'derivative' (or 'power'), as the derivative of a variable is not zero.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 applies both the sum rule and the constant-multiple rule simultaneously, violating the one-rule-per-step constraint. Step 4 incorrectly labels the differentiation of x as 'constant'; it should be 'derivative' (or 'power'), as the derivative of a variable is not zero.gpt-oss:20b: fail (error) 2026-10-03 — Step 1 applies the derivative rule directly to the product 2*x*(x‑1) instead of using the product rule, so the first step is incorrect. Step 4 then incorrectly treats the derivative of x as a constant, yielding 0; the correct derivative is 1. These errors invalidate the solution.qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 4 claims to apply the 'constant' rule, but it actually applies the power rule to differentiate x (Derivative(x, x) -> 1). The 'constant' rule applies to derivatives of constant values, not to the variable x.gpt-oss:20b: fail (style) 2026-09-30 — Step 4 labels the use of the derivative of x as "constant"; the correct rule name is "derivative".
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 2026-10-03 with SymPy 1.14.0.