Derivative of \( \displaystyle 2 x \left(25 x^{2} + 15 x + 3\right) \)
Problem 2.1087 · hard Video
Differentiate \( \displaystyle f(x) = 2 x \left(25 x^{2} + 15 x + 3\right) \).
- \[ \frac{d}{d x} 2 x \left(25 x^{2} + 15 x + 3\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 2 x \frac{d}{d x} \left(25 x^{2} + 15 x + 3\right) + \left(25 x^{2} + 15 x + 3\right) \frac{d}{d x} 2 x \]productApply the product rule.✓ Proved
- \[ = 50 x^{2} + 2 x \frac{d}{d x} \left(25 x^{2} + 15 x + 3\right) + 30 x + 6 \]constant-multipleFactor out the constant 2 from the first term.✓ Proved
- \[ = 50 x^{2} + 2 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 15 x + \frac{d}{d x} 25 x^{2}\right) + 30 x + 6 \]sumApply the sum rule to the second term.✓ Proved
- \[ = 50 x^{2} + 2 x \left(50 x + 15\right) + 30 x + 6 \]derivative simplifyDifferentiate each term inside the parentheses. Simplify the expression by removing the zero.✓ Proved
- \[ = 150 x^{2} + 60 x + 6 \]algebra simplifyDistribute the terms. Combine like terms.✓ Proved
Answer \( 6 \left(5 x + 1\right)^{2} \)
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 |
| 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 (error) — Step 3 incorrectly factors out the constant 2 from the first term, dropping the factor of x and changing the expression from 2*x*(25*x**2+15*x+3) to 2*(25*x**2+15*x+3). This applies more than one rule at once and results in an algebraically incorrect intermediate expression.qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, sum rule, and power rule in distinct steps. The final simplification is algebraically correct and matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the product rule, sum rule, and power rule in distinct steps. The final simplification is algebraically correct and matches the stated answer.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 incorrectly factors out the constant 2 from the first term, dropping the factor of x and changing the expression from 2*x*(25*x**2+15*x+3) to 2*(25*x**2+15*x+3). This applies more than one rule at once and results in an algebraically incorrect intermediate expression.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: fail (style) 2026-09-28 — Step 7 distributes both products in one line, applying two algebraic expansions at once, violating the rule that each step must change only one thing.
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 2026-09-29 with SymPy 1.14.0.