Derivative of \( \displaystyle 32 x^{3} + 48 x^{2} + 24 x \)
Problem 2.1841 · medium
Differentiate \( \displaystyle f(x) = 32 x^{3} + 48 x^{2} + 24 x \).
- \[ \frac{d}{d x} \left(32 x^{3} + 48 x^{2} + 24 x\right) \]sumStart with the derivative of the entire function.✓ Proved
- \[ = \frac{d}{d x} 24 x + \frac{d}{d x} 48 x^{2} + \frac{d}{d x} 32 x^{3} \]sumApply the sum rule to differentiate each term separately.✓ Proved
- \[ = 24 \frac{d}{d x} x + 48 \frac{d}{d x} x^{2} + 32 \frac{d}{d x} x^{3} \]constant-multipleFactor out the constants from each derivative.✓ Proved
- \[ = 96 x^{2} + 96 x + 24 \]power algebraApply the power rule to each term. Multiply the constants.✓ Proved
Answer \( 24 \left(2 x + 1\right)^{2} \)
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.
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) — Step 3 contains a typographical error: "2_4*Derivative(x, x)" should be "24*Derivative(x, x)". This mislabels the constant and would lead to an incorrect intermediate expression if not corrected.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08gpt-oss:20b: fail (error) 2026-10-08 — Step 3 contains a typographical error: "2_4*Derivative(x, x)" should be "24*Derivative(x, x)". This mislabels the constant and would lead to an incorrect intermediate expression if not corrected.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.