Derivative of \( \displaystyle x \left(- x - 4\right) \)
Problem 2.1379 · medium
Differentiate \( \displaystyle f(x) = x \left(- x - 4\right) \).
- \[ \frac{d}{d x} x \left(- x - 4\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(- x - 4\right) + \left(- x - 4\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = x \frac{d}{d x} \left(- x - 4\right) - x - 4 \]constant-multipleDifferentiate the first part of the product.✓ Proved
- \[ = x \frac{d}{d x} \left(-4\right) + x \frac{d}{d x} \left(- x\right) - x - 4 \]sumApply the sum rule to the second part.✓ Proved
- \[ = - 2 x - 4 \]derivative constant-multiple algebraDifferentiate the individual terms. Simplify the terms. Combine like terms.✓ Proved
Answer \( - 2 x - 4 \)
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.
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 |
| 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: fail (error) — Step 3 is labeled 'constant-multiple' but performs the differentiation of x to 1, which is a 'derivative' step. Step 4 is labeled 'sum' but distributes x over the sum inside the derivative, which is an algebraic distribution or linearity step, not the sum rule for differentiation (which would split Derivative(A+B) into Derivative(A)+Derivative(B)).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is labeled 'constant-multiple' but performs the differentiation of x to 1, which is a 'derivative' step. Step 4 is labeled 'sum' but distributes x over the sum inside the derivative, which is an algebraic distribution or linearity step, not the sum rule for differentiation (which would split Derivative(A+B) into Derivative(A)+Derivative(B)).gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is labeled 'constant-multiple' but performs the differentiation of x to 1, which is a 'derivative' step. Step 4 is labeled 'sum' but distributes x over the sum inside the derivative, which is an algebraic expansion or linearity step, not the sum rule for derivatives (which would split Derivative(A+B) into Derivative(A)+Derivative(B)).gpt-oss:20b: fail (error) 2026-10-03 — Step 6 combines two simplifications—applying constant‑multiple to both x*(-1) and x*0 and then summing the terms—into a single step, violating the rule that each step must change only one thing. The label "constant‑multiple" does not cover the removal of the zero term, so the step is incorrectly labeled and over‑applies rules.
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.