Derivative of \( \displaystyle 5 x \left(16 x^{2} + 12 x + 3\right) \)
Problem 2.513 · hard Video
Differentiate \( \displaystyle f(x) = 5 x \left(16 x^{2} + 12 x + 3\right) \).
- \[ \frac{d}{d x} 5 x \left(16 x^{2} + 12 x + 3\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 5 \frac{d}{d x} x \left(16 x^{2} + 12 x + 3\right) \]constant-multiplePull out the constant factor 5.✓ Proved
- \[ = 5 x \frac{d}{d x} \left(16 x^{2} + 12 x + 3\right) + 5 \left(16 x^{2} + 12 x + 3\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 80 x^{2} + 5 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 12 x + \frac{d}{d x} 16 x^{2}\right) + 60 x + 15 \]simplifySimplify the derivative of the first term.✓ Proved
- \[ = 80 x^{2} + 5 x \left(32 x + 12\right) + 60 x + 15 \]derivative simplifyDifferentiate each term inside the parentheses. Simplify the derivative of the constant term.✓ Proved
- \[ = 240 x^{2} + 120 x + 15 \]algebra simplify simplifyDistribute x into the parentheses. Combine like terms inside the parentheses. Distribute the 5 to get the final simplified expression.✓ Proved
Answer \( 15 \left(4 x + 1\right)^{2} \)
✓ 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. 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: fail (style) — Step 4 is mislabeled: it applies the derivative rule to Derivative(x,x) but is marked as "simplify". The label should be "derivative" (or "simplify" only if no differentiation is performed).qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final result is algebraically equivalent to the stated answer.
Senior review claude-sonnet-5-5, 2026-10-09: fail — The math is correct: the derivative is 240x^2+120x+15 = 15(4x+1)^2, and every line is proved equivalent. Step 4 combines evaluating d/dx[x] with the sum-rule split and is labeled 'simplify', which violates the one-change-per-step contract. It should be split into two steps or labeled accurately.
gpt-oss:20b: uphold — Line 3 to line 4 evaluates Derivative(x,x) to 1 and also splits the derivative of the sum into three terms (sum rule); that is two independent changes, and the 'simplify' label names neither a derivative evaluation nor the sum rule.
Every verdict on record (7)
claude-sonnet-5-5: fail (style) 2026-10-09 — The math is correct: the derivative is 240x^2+120x+15 = 15(4x+1)^2, and every line is proved equivalent. Step 4 combines evaluating d/dx[x] with the sum-rule split and is labeled 'simplify', which violates the one-change-per-step contract. It should be split into two steps or labeled accurately.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: fail (style) 2026-09-21 — Step 4 is mislabeled: it applies the derivative rule to Derivative(x,x) but is marked as "simplify". The label should be "derivative" (or "simplify" only if no differentiation is performed).qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the final result is algebraically equivalent to the stated answer.gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies a derivative to the sum 16*x**2+12*x+3 but labels the rule as "simplify". It should be labeled "derivative" (or a combination of "derivative" and "sum"), so the step violates the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 4 applies the linearity of the derivative (sum rule) to split the derivative of the second factor, but labels this step 'simplify'. The label 'simplify' is reserved for presentational algebraic simplification, not for applying differentiation rules.gpt-oss:20b: pass 2026-09-20
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.