Derivative of \( \displaystyle - x^{3} - 6 x^{2} - 12 x \)
Problem 2.2443 · medium Video
Differentiate \( \displaystyle f(x) = - x^{3} - 6 x^{2} - 12 x \).
- \[ \frac{d}{d x} \left(- x^{3} - 6 x^{2} - 12 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 12 x\right) + \frac{d}{d x} \left(- 6 x^{2}\right) + \frac{d}{d x} \left(- x^{3}\right) \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 6 x^{2}\right) + \frac{d}{d x} \left(- x^{3}\right) - 12 \]derivativeDifferentiate the last term.✓ Proved
- \[ = - 6 \frac{d}{d x} x^{2} + \frac{d}{d x} \left(- x^{3}\right) - 12 \]constant-multipleApply the constant multiple rule to the second term.✓ Proved
- \[ = - 12 x + \frac{d}{d x} \left(- x^{3}\right) - 12 \]powerApply the power rule to x**2.✓ Proved
- \[ = - 3 x^{2} - 12 x - 12 \]simplifySimplify the entire expression.✓ Proved
Answer \( - 3 \left(x + 2\right)^{2} \)
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 |
| 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 2026-10-19gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 6 applies the derivative rule to the first term (-x**3) and simplifies the second term (-6*2*x) simultaneously, violating the one-change-per-step constraint. Additionally, the label 'simplify' is incorrect for a step that performs differentiation.gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.