Derivative of \( \displaystyle x \left(6 - x\right) \)
Problem 2.1078 · medium
Differentiate \( \displaystyle f(x) = x \left(6 - x\right) \).
- \[ \frac{d}{d x} x \left(6 - x\right) \]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
- \[ = 6 - \frac{d}{d x} x^{2} \]constantThe derivative of 6*x with respect to x is 6.✓ Proved
- \[ = 6 - 2 x \]powerApply the power rule to x**2.✓ Proved
Answer \( 6 - 2 x \)
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 |
| 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: passqwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'constant', but the operation performed is the derivative of a linear term (6x), which is better described by 'constant-multiple' or 'power'. The label 'constant' typically refers to the derivative of a constant being zero, which is not what happens here.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 4 is labeled 'constant', but the operation performed is the derivative of a linear term (6x), which is better described by 'constant-multiple' or 'power'. The label 'constant' typically refers to the derivative of a constant being zero, which is not what happens here.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 4 applies two rules at once: it uses the constant-multiple rule to handle the coefficient 6 and the power rule to differentiate x, violating the one-rule-per-step constraint. Additionally, the label 'constant' is incorrect for differentiating 6*x; it should be 'constant-multiple' or 'power'.gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.