Derivative of \( \displaystyle x \left(- 3 x - 4\right) \)
Problem 2.1216 · medium Video
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. 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: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-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 2026-10-03 with SymPy 1.14.0.