Derivative of \( \displaystyle 2 x \left(16 x^{2} - 12 x + 3\right) \)
Problem 2.1324 · hard
Differentiate \( \displaystyle f(x) = 2 x \left(16 x^{2} - 12 x + 3\right) \).
- \[ \frac{d}{d x} 2 x \left(16 x^{2} - 12 x + 3\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 2 x \frac{d}{d x} \left(16 x^{2} - 12 x + 3\right) + \left(16 x^{2} - 12 x + 3\right) \frac{d}{d x} 2 x \]productApply the product rule.✓ Proved
- \[ = 32 x^{2} + 2 x \frac{d}{d x} \left(16 x^{2} - 12 x + 3\right) - 24 x + 6 \]constant-multipleDifferentiate the first part of the product.✓ Proved
- \[ = 32 x^{2} + 2 x \left(\frac{d}{d x} 3 - \frac{d}{d x} 12 x + \frac{d}{d x} 16 x^{2}\right) - 24 x + 6 \]sumDistribute the derivative across the sum.✓ Proved
- \[ = 32 x^{2} + 2 x \left(32 x - 12\right) - 24 x + 6 \]derivative simplifyDifferentiate each term in the sum. Simplify the expression.✓ Proved
- \[ = 96 x^{2} - 48 x + 6 \]algebra simplifyDistribute the terms. Combine like terms.✓ Proved
Answer \( 6 \left(4 x - 1\right)^{2} \)
✓ 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.
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 |
| 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: fail (style) — Step 3 applies the derivative rule to 2*x, but labels it as constant‑multiple. The correct label should be "derivative".qwen3.6:27b-mlx: pass — 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.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — 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 (style) 2026-10-03 — Step 3 applies the derivative rule to 2*x, but labels it as constant‑multiple. The correct label should be "derivative".qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies the product rule, linearity of differentiation, and basic derivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.