Derivative of \( \displaystyle x \left(x - 6\right) \)
Problem 2.278 · easy
Differentiate \( \displaystyle f(x) = x \left(x - 6\right) \).
- \[ \frac{d}{d x} x \left(x - 6\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(x^{2} - 6 x\right) \]algebraExpand the expression.✓ Proved
- \[ = - \frac{d}{d x} 6 x + \frac{d}{d x} x^{2} \]sumApply the difference rule.✓ Proved
- \[ = 2 x - \frac{d}{d x} 6 x \]powerDifferentiate the first term using the power rule.✓ Proved
- \[ = 2 x - 6 \]constant-multipleDifferentiate the second term by pulling out the constant.✓ Proved
Answer \( 2 x - 6 \)
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 |
| 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: fail (error) — Step 5 is mathematically incorrect because it claims to apply the constant-multiple rule to obtain the final result, but the constant-multiple rule only factors out the constant (yielding 6 * Derivative(x, x)); it does not perform the differentiation of x to 1. The step conflates factoring with differentiation.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 is mathematically incorrect because it claims to apply the constant-multiple rule to obtain the final result, but the constant-multiple rule only factors out the constant (yielding 6 * Derivative(x, x)); it does not perform the differentiation of x to 1. The step conflates factoring with differentiation.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies one rule per step, uses valid labels from the fixed vocabulary, and the notes are accurate.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 is labeled 'constant-multiple', but the step actually performs the differentiation of x (derivative rule) and simplification. The constant-multiple rule would only pull the 6 out, leaving Derivative(x, x), not the final result 6.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-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.