Derivative of \( \displaystyle 6 x \left(x - 1\right) \)
Problem 2.201 · easy
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 \( 12 x - 6 \)
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: 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.
Every verdict on record (15)
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.