Derivative of \( \displaystyle 2 x \left(1 - x\right) \)
Problem 2.55 · easy
Differentiate \( \displaystyle f(x) = 2 x \left(1 - x\right) \).
- \[ \frac{d}{d x} 2 x \left(1 - x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 2 \frac{d}{d x} x \left(1 - x\right) \]constant-multiplePull out the constant factor 2.✓ Proved
- \[ = 2 x \frac{d}{d x} \left(1 - x\right) + 2 \left(1 - x\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 2 x \frac{d}{d x} \left(1 - x\right) - 2 x + 2 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = 2 x \frac{d}{d x} 1 - 2 x \frac{d}{d x} x - 2 x + 2 \]algebraDistribute the derivative over the subtraction.✓ Proved
- \[ = 2 - 4 x \]derivative simplify algebra algebraDifferentiate the remaining terms. Simplify the expression inside the parentheses. Combine like terms. Distribute the 2.✓ Proved
Answer \( 2 - 4 x \)
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 |
| 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed.gpt-oss:20b: fail (error) 2026-09-20 — Step 6 applies two derivative rules at once—derivative of 1 and derivative of x—violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple, product, and derivative rules in a step-by-step manner. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 applies the linearity of the derivative operator (distributing the derivative over subtraction) but labels it 'algebra'. The label 'algebra' is reserved for presentational steps with no differentiation; this step performs a differentiation operation, so it should be labeled 'sum' (or 'derivative' if linearity is subsumed, but 'sum' is the standard label for linearity in this vocabulary). Furthermore, Step 5 splits Derivative(1-x, x) into Derivative(1, x) - Derivative(x, x) in one go, which is applying the sum rule and then preparing for derivative evaluation, but the label 'algebra' is definitively wrong for a step that manipulates derivative operators.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 5 should be labeled 'derivative' instead of 'algebra' as it involves computing derivatives.gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 5 incorrectly labels the application of the linearity of the derivative as an algebraic step; the note and rule do not match the actual operation performed.deepseek-r1:70b: pass 2026-09-17
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 2026-09-26 with SymPy 1.14.0.