Derivative of \( \displaystyle x \left(4 x^{2} - 6 x + 3\right) \)
Problem 2.979 · hard Video
Differentiate \( \displaystyle f(x) = x \left(4 x^{2} - 6 x + 3\right) \).
- \[ \frac{d}{d x} x \left(4 x^{2} - 6 x + 3\right) \]Start with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(4 x^{2} - 6 x + 3\right) + \left(4 x^{2} - 6 x + 3\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 4 x^{2} + x \frac{d}{d x} \left(4 x^{2} - 6 x + 3\right) - 6 x + 3 \]derivative constant-multipleDifferentiate the first part of the product. Simplify the first term.✓ Proved
- \[ = 4 x^{2} + x \left(\frac{d}{d x} 3 - \frac{d}{d x} 6 x + \frac{d}{d x} 4 x^{2}\right) - 6 x + 3 \]sumDistribute the derivative over the sum.✓ Proved
- \[ = 4 x^{2} + x \left(8 x - 6\right) - 6 x + 3 \]derivative simplifyDifferentiate each term in the sum. Simplify the derivative terms.✓ Proved
- \[ = 12 x^{2} - 12 x + 3 \]product simplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( 3 \left(2 x - 1\right)^{2} \)
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. 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: passqwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'constant-multiple' but performs algebraic simplification (multiplying by 1); the label should be 'simplify' or 'algebra'. Step 8 is labeled 'product' but performs algebraic distribution; the label should be 'algebra' or 'simplify'.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 4 is labeled 'constant-multiple' but performs algebraic simplification (multiplying by 1); the label should be 'simplify' or 'algebra'. Step 8 is labeled 'product' but performs algebraic distribution; the label should be 'algebra' or 'simplify'.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the product rule, linearity of differentiation, and power rule. Each step changes only one aspect of the expression, and the labels accurately reflect the rules applied.gpt-oss:20b: fail (error) 2026-09-27 — Step 5 applies two rules at once: it both distributes the derivative over a sum (sum rule) and applies the derivative to each term. Each step must change only one thing, so this is a defect.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the product rule, linearity of differentiation, and power rule. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-27
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.