Derivative of \( \displaystyle x \left(25 x^{2} - 15 x + 3\right) \)
Problem 2.716 · hard Video
Differentiate \( \displaystyle f(x) = x \left(25 x^{2} - 15 x + 3\right) \).
- \[ \frac{d}{d x} x \left(25 x^{2} - 15 x + 3\right) \]Start with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(25 x^{2} - 15 x + 3\right) + \left(25 x^{2} - 15 x + 3\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 25 x^{2} + x \left(\frac{d}{d x} 3 - \frac{d}{d x} 15 x + \frac{d}{d x} 25 x^{2}\right) - 15 x + 3 \]constantDifferentiate the first part of the product.✓ Proved
- \[ = 25 x^{2} + x \left(\frac{d}{d x} 25 x^{2} - 15\right) - 15 x + 3 \]derivative algebraDifferentiate the terms in the second part. Simplify the terms inside the parentheses.✓ Proved
- \[ = 25 x^{2} + x \left(50 x - 15\right) - 15 x + 3 \]derivative constantDifferentiate the power term. Simplify the first part of the expression.✓ Proved
- \[ = 75 x^{2} - 30 x + 3 \]algebra simplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( 3 \left(5 x - 1\right)^{2} \)
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. A reviewer's objection to the explanation stands; see 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: fail (style) — Step 3 incorrectly labels the operation as "constant" while actually applying the derivative (sum rule) to each term inside the parentheses. The label should be "derivative" or "sum".qwen3.6:27b-mlx: fail (error) — Step 3 applies multiple rules simultaneously: it differentiates the first term of the product (derivative rule), distributes the derivative operator over the sum inside the second term (sum rule), and simplifies the coefficient 1 (algebra/simplify). This violates the constraint that each step must change only one thing.
Senior review claude-sonnet-5-5, 2026-10-12: fail — The math is correct throughout, but step 3 does two independent things (differentiates x, and splits the derivative of the second factor over the sum) under the wrong label 'constant', which would mislead a student. Step 7 is labelled 'constant' for dropping a factor of 1, which is acceptable but loose.
gpt-oss:20b: uphold — Step 3 turns d/dx[x] into 1 and splits the sum in the second factor, and neither action is the 'constant' rule; the step should be labelled power, derivative or sum.qwen3.6:27b-mlx: uphold — The step does two independent things, evaluating d/dx[x]=1 in the first factor and applying the sum rule to the second factor. It does not simplify a coefficient, so that part of the objection is wrong, but the step still bundles two changes.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-12 — The math is correct throughout, but step 3 does two independent things (differentiates x, and splits the derivative of the second factor over the sum) under the wrong label 'constant', which would mislead a student. Step 7 is labelled 'constant' for dropping a factor of 1, which is acceptable but loose.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 applies multiple rules simultaneously: it differentiates the first term of the product (derivative rule), distributes the derivative operator over the sum inside the second term (sum rule), and simplifies the coefficient 1 (algebra/simplify). This violates the constraint that each step must change only one thing.gpt-oss:20b: fail (style) 2026-09-26 — Step 3 incorrectly labels the operation as "constant" while actually applying the derivative (sum rule) to each term inside the parentheses. The label should be "derivative" or "sum".qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 applies multiple rules simultaneously: it differentiates the first term of the product (derivative), distributes the derivative operator over the sum inside the second term (sum/linearity), and simplifies the coefficient 1 (algebra). This violates the 'one rule per step' constraint.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the application of the sum rule as "constant". The step applies the sum rule to differentiate 25*x**2 - 15*x + 3, not the constant rule.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 applies multiple rules simultaneously: it differentiates the first term of the product (derivative rule), distributes the derivative over the sum inside the second term (sum rule), and simplifies the coefficient of the first term (algebra/simplify). This violates the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies the derivative to two terms (15*x and 3) in a single line and labels the operation as "constant", which is incorrect. This violates the rule that each step must change only one thing and use the correct label.
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.