Derivative of \( \displaystyle - 16 x^{3} - 12 x^{2} - 3 x \)
Problem 2.2439 · medium Video
Differentiate \( \displaystyle f(x) = - 16 x^{3} - 12 x^{2} - 3 x \).
- \[ \frac{d}{d x} \left(- 16 x^{3} - 12 x^{2} - 3 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(- 12 x^{2}\right) + \frac{d}{d x} \left(- 16 x^{3}\right) \]sumApply the sum rule to differentiate each term separately.✓ Proved
- \[ = - 3 \frac{d}{d x} x - 12 \frac{d}{d x} x^{2} - 16 \frac{d}{d x} x^{3} \]constant-multiplePull out the constant coefficients.✓ Proved
- \[ = - 48 x^{2} - 24 x - 3 \]power algebraApply the power rule to each term. Multiply the constants.✓ Proved
Answer \( - 3 \left(4 x + 1\right)^{2} \)
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 (error) — The final answer -3*(4*x + 1)**2 does not equal the computed derivative -48*x**2 - 24*x - 3. The solution’s steps are correct, but the final result is incorrect.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19gpt-oss:20b: fail (error) 2026-10-19 — The final answer -3*(4*x + 1)**2 does not equal the computed derivative -48*x**2 - 24*x - 3. The solution’s steps are correct, but the final result is incorrect.qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: fail (error) 2026-10-17 — The final answer given (-3*(4*x + 1)**2) does not match the result obtained from the differentiation steps (-48*x**2 - 24*x - 3). The solution is mathematically incorrect.qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: fail (error) 2026-10-17 — The final stated answer -3*(4*x + 1)**2 does not match the derivative obtained in the steps, which is -48*x**2 - 24*x - 3. The solution incorrectly claims the two expressions are equivalent.
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.