Derivative of \( \displaystyle 6 x \left(x - 1\right) \)
Problem 2.201 · easy Video
Differentiate \( \displaystyle f(x) = 6 x \left(x - 1\right) \).
- \[ \frac{d}{d x} 6 x \left(x - 1\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 6 \frac{d}{d x} x \left(x - 1\right) \]constant-multiplePull out the constant 6.✓ Proved
- \[ = 6 x \frac{d}{d x} \left(x - 1\right) + 6 \left(x - 1\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 6 x \frac{d}{d x} \left(x - 1\right) + 6 x - 6 \]derivative algebraDifferentiate x and (x - 1). Simplify the term 1*(x - 1).✓ Proved
- \[ = 12 x - 6 \]derivative algebra algebra algebraThe derivative of x - 1 is 1. Substitute the derivative value. Combine like terms inside the parentheses. Distribute the 6.✓ Proved
Answer \( 12 x - 6 \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify
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: fail (error) — Step 4 is labeled 'derivative' but applies the product rule to the term x*(x-1) while also differentiating x. The product rule was already applied in Step 3; Step 4 should only be evaluating the derivatives of the factors (derivative of x and derivative of x-1), but the label 'derivative' is insufficient to describe the structural change if it implies re-applying product rule, or it is a multi-rule step if it simplifies 1*(x-1) as well. More critically, Step 4 claims to differentiate 'x and (x-1)' but the expression still contains 'Derivative(x - 1, x)', meaning it only differentiated x. The label 'derivative' is acceptable for differentiating x, but the note says it differentiates both, which is false since Derivative(x-1, x) remains. This is a misleading note. However, looking closer at the 'one rule per step' constraint: Step 4 changes Derivative(x, x) to 1. This is a 'derivative' step. It leaves Derivative(x-1, x) alone. This is valid. But wait, Step 4's note says 'Differentiate x and (x - 1)'. It did NOT differentiate (x-1) in this step; that happens in Step 6. The note is factually false about what the step did. This is a 'misleading' defect. Is it an error? The math is correct, the label is correct for the change made (differentiating x). The note is wrong. Let's look for a harder defect. Step 4: `6*(Derivative(x, x)*(x - 1) + x*Derivative(x - 1, x))` becomes `6*(1*(x - 1) + x*Derivative(x - 1, x))`. The change is `Derivative(x, x)` -> `1`. Rule: `derivative`. This is correct. Note: 'Differentiate x and (x - 1)'. This is false, only x was differentiated. This is misleading. Step 6: `6*(x - 1 + x*Derivative(x - 1, x))` becomes `6*(x - 1 + x*1)`. Change: `Derivative(x - 1, x)` -> `1`. Rule: `derivative`. Correct. Note: 'The derivative of x - 1 is 1'. Correct. So the main issue is the note in Step 4. Is there a rule violation? Step 4 applies one rule (derivative of x). Step 5 applies algebra (1*(x-1) -> x-1). Step 6 applies derivative (derivative of x-1). This seems compliant with granularity. The note is the only flaw. 'Misleading' severity.
Senior review claude-sonnet-5-5, 2026-10-06: pass — All lines are SymPy-proved, rule labels are legitimate, each step changes one thing, and the final line matches the stated answer 12*x - 6. The only blemish is slightly loose note wording in steps 4 and 7, which is not a defect under the contract.
qwen3.6:27b-mlx: dismiss — The step correctly changes Derivative(x, x) to 1 under the 'derivative' label and SymPy proved it; the note's wording ('x and x-1') is loose, but the contract judges labels and math, not note precision, and no false statement or wrong rule name results. Step 7's note ('Substitute the derivative value') is also imprecise, but that is style at most.
Every verdict on record (16)
claude-sonnet-5-5: pass 2026-10-06 — All lines are SymPy-proved, rule labels are legitimate, each step changes one thing, and the final line matches the stated answer 12*x - 6. The only blemish is slightly loose note wording in steps 4 and 7, which is not a defect under the contract.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 4 is labeled 'derivative' but applies the product rule to the term x*(x-1) while also differentiating x. The product rule was already applied in Step 3; Step 4 should only be evaluating the derivatives of the factors (derivative of x and derivative of x-1), but the label 'derivative' is insufficient to describe the structural change if it implies re-applying product rule, or it is a multi-rule step if it simplifies 1*(x-1) as well. More critically, Step 4 claims to differentiate 'x and (x-1)' but the expression still contains 'Derivative(x - 1, x)', meaning it only differentiated x. The label 'derivative' is acceptable for differentiating x, but the note says it differentiates both, which is false since Derivative(x-1, x) remains. This is a misleading note. However, looking closer at the 'one rule per step' constraint: Step 4 changes Derivative(x, x) to 1. This is a 'derivative' step. It leaves Derivative(x-1, x) alone. This is valid. But wait, Step 4's note says 'Differentiate x and (x - 1)'. It did NOT differentiate (x-1) in this step; that happens in Step 6. The note is factually false about what the step did. This is a 'misleading' defect. Is it an error? The math is correct, the label is correct for the change made (differentiating x). The note is wrong. Let's look for a harder defect. Step 4: `6*(Derivative(x, x)*(x - 1) + x*Derivative(x - 1, x))` becomes `6*(1*(x - 1) + x*Derivative(x - 1, x))`. The change is `Derivative(x, x)` -> `1`. Rule: `derivative`. This is correct. Note: 'Differentiate x and (x - 1)'. This is false, only x was differentiated. This is misleading. Step 6: `6*(x - 1 + x*Derivative(x - 1, x))` becomes `6*(x - 1 + x*1)`. Change: `Derivative(x - 1, x)` -> `1`. Rule: `derivative`. Correct. Note: 'The derivative of x - 1 is 1'. Correct. So the main issue is the note in Step 4. Is there a rule violation? Step 4 applies one rule (derivative of x). Step 5 applies algebra (1*(x-1) -> x-1). Step 6 applies derivative (derivative of x-1). This seems compliant with granularity. The note is the only flaw. 'Misleading' severity.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 4 is labeled 'derivative' but applies the product rule to x*(x-1) while also differentiating x. The product rule application should have been labeled 'product' (or the step should have been split). Furthermore, Step 4 claims to differentiate both x and (x-1) in one step, violating the one-rule-per-step constraint by combining the product rule expansion with the evaluation of d/dx(x). Specifically, the transition from step 3 to 4 applies the product rule structure but also evaluates d/dx(x)=1, which is two operations. A cleaner split would be: apply product rule to get 6*(Derivative(x,x)*(x-1) + x*Derivative(x-1,x)) [product], then evaluate Derivative(x,x) to 1 [derivative], then evaluate Derivative(x-1,x) to 1 [derivative]. Step 4 does both evaluations and the structural application of the product rule's result in a messy way. Actually, looking closer: Step 3 is `6*(Derivative(x, x)*(x - 1) + x*Derivative(x - 1, x))`. Step 4 is `6*(1*(x - 1) + x*Derivative(x - 1, x))`. This step changes `Derivative(x, x)` to `1`. This is a single application of the derivative rule for x. The label 'derivative' is correct for this specific change. However, the note says 'Differentiate x and (x - 1)'. The step only differentiates x. It leaves `Derivative(x - 1, x)` untouched. So the note is misleading/false because it claims to differentiate (x-1) in this step, but it doesn't. Step 6 handles (x-1). So the note in step 4 is misleading.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 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 4 is labeled 'derivative' but performs two distinct operations: differentiating x to get 1 and applying the product rule structure. More critically, the note claims it differentiates both x and (x-1), but the expression still contains Derivative(x-1, x), meaning it only differentiated x. This violates the one-change-per-step rule and mislabels the action.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-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.