Derivative of \( \displaystyle 2 x \left(4 x^{2} - 18 x + 27\right) \)
Problem 2.1000 · hard Video
Differentiate \( \displaystyle f(x) = 2 x \left(4 x^{2} - 18 x + 27\right) \).
- \[ \frac{d}{d x} 2 x \left(4 x^{2} - 18 x + 27\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(8 x^{3} - 36 x^{2} + 54 x\right) \]algebraDistribute the 2*x into the parentheses.✓ Proved
- \[ = 54 \frac{d}{d x} x - 36 \frac{d}{d x} x^{2} + 8 \frac{d}{d x} x^{3} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = 24 x^{2} - 72 x + 54 \]power simplifyApply the power rule to each term. Simplify the resulting expression.✓ Proved
Answer \( 6 \left(2 x - 3\right)^{2} \)
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 |
| 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 1 is missing a rule label; it should be labeled "derivative" to indicate the application of the derivative rule.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: fail (style) 2026-09-28 — Step 1 is missing a rule label; it should be labeled "derivative" to indicate the application of the derivative rule.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly expands the polynomial, applies the linearity of the derivative (sum rule), and uses the power rule. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-27 — Step 1 does not apply any rule; it merely restates the derivative. The product rule is missing, so the solution skips a required step and violates the one‑change‑per‑step rule.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27
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 2026-09-28 with SymPy 1.14.0.