Derivative of \( \displaystyle x \left(- 3 x - 4\right) \)
Problem 2.1216 · medium
Differentiate \( \displaystyle f(x) = x \left(- 3 x - 4\right) \).
- \[ \frac{d}{d x} x \left(- 3 x - 4\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(- 3 x - 4\right) + \left(- 3 x - 4\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = x \frac{d}{d x} \left(- 3 x - 4\right) - 3 x - 4 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = x \frac{d}{d x} \left(-4\right) + x \frac{d}{d x} \left(- 3 x\right) - 3 x - 4 \]sumApply the sum rule to the second part.✓ Proved
- \[ = - 3 x \frac{d}{d x} x - 3 x - 4 \]constant-multipleDifferentiate the terms in the sum.✓ Proved
- \[ = - 6 x - 4 \]derivative algebra simplifyDifferentiate the remaining x terms. Simplify the expression. Combine like terms.✓ Proved
Answer \( - 6 x - 4 \)
✓ 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 6 incorrectly labels a simplification step as "derivative"; no differentiation occurs there.qwen3.6:27b-mlx: pass — The solution correctly applies the product rule and sum rule, with appropriate labels for each step. The algebraic simplifications are handled in separate steps as required.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the product rule and sum rule, with appropriate labels for each step. The algebraic simplifications are handled in separate steps as required.gpt-oss:20b: fail (style) 2026-09-29 — Step 6 incorrectly labels a simplification step as "derivative"; no differentiation occurs there.qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 4 applies the sum rule to the argument of the derivative but also distributes the factor 'x' across the resulting terms in a single step, violating the one-rule-per-step constraint. Step 5 applies constant-multiple and derivative rules simultaneously to multiple terms.gpt-oss:20b: pass 2026-09-29
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.