Derivative of \( \displaystyle 2 x \left(x - 3\right) \)
Problem 2.1008 · medium
Differentiate \( \displaystyle f(x) = 2 x \left(x - 3\right) \).
- \[ \frac{d}{d x} 2 x \left(x - 3\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(2 x^{2} - 6 x\right) \]algebraExpand the expression first.✓ Proved
- \[ = - 6 \frac{d}{d x} x + 2 \frac{d}{d x} x^{2} \]sumApply the sum rule.✓ Proved
- \[ = 2 \frac{d}{d x} x^{2} - 6 \]constantThe derivative of a constant is zero.✓ Proved
- \[ = 4 x - 6 \]power simplifyApply the power rule to x**2. Simplify the final expression.✓ Proved
Answer \( 4 x - 6 \)
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 (misleading) — Step 4 replaces –6·Derivative(x,x) with –6 using the label ‘constant’, but it actually applies the derivative rule for x (Derivative(x,x)=1) and the constant‑multiple rule. This step applies two rules at once and mislabels the operation.qwen3.6:27b-mlx: fail (style) — 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'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-27 — 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'.gpt-oss:20b: fail (misleading) 2026-09-27 — Step 4 replaces –6·Derivative(x,x) with –6 using the label ‘constant’, but it actually applies the derivative rule for x (Derivative(x,x)=1) and the constant‑multiple rule. This step applies two rules at once and mislabels the operation.qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 3 applies both the sum rule and the constant-multiple rule simultaneously, violating the one-change-per-step constraint. Step 4 is labeled 'constant' but actually applies the derivative of x (a power rule case) and removes the derivative operator, which is better described as 'derivative' or 'power'.gpt-oss:20b: fail (style) 2026-09-27 — Step 4 incorrectly labels the application of the derivative of x as a "constant" rule. The correct label should be "derivative" since the derivative of x is 1, not a constant. The mathematical steps themselves are correct, but the label misrepresents the rule used.
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 2026-09-27 with SymPy 1.14.0.