Derivative of \( \displaystyle 18 x^{3} + 36 x^{2} + 24 x \)
Problem 2.1494 · medium
Differentiate \( \displaystyle f(x) = 18 x^{3} + 36 x^{2} + 24 x \).
- \[ \frac{d}{d x} \left(18 x^{3} + 36 x^{2} + 24 x\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 24 x + \frac{d}{d x} 36 x^{2} + \frac{d}{d x} 18 x^{3} \]sumApply the sum rule.✓ Proved
- \[ = 2 \frac{d}{d x} 12 x + 36 \frac{d}{d x} x^{2} + 18 \frac{d}{d x} x^{3} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 24 \frac{d}{d x} x + 36 \frac{d}{d x} x^{2} + 18 \frac{d}{d x} x^{3} \]constant-multipleSimplify the constant terms.✓ Proved
- \[ = 54 x^{2} + 72 x + 24 \]power algebraApply the power rule to each term. Multiply the constants.✓ Proved
Answer \( 6 \left(3 x + 2\right)^{2} \)
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.
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: fail (error) — Step 3 applies two rules at once: it rewrites 24*Derivative(x,x) as 2*Derivative(12*x,x), which both changes the constant factor and rewrites the inner expression. This violates the one‑rule‑per‑step rule and also misstates the derivative of 24*x.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: fail (error) 2026-10-04 — Step 3 applies two rules at once: it rewrites 24*Derivative(x,x) as 2*Derivative(12*x,x), which both changes the constant factor and rewrites the inner expression. This violates the one‑rule‑per‑step rule and also misstates the derivative of 24*x.qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 3 incorrectly applies the constant-multiple rule to the term 24*x by rewriting it as 2*Derivative(12*x, x), which changes the internal structure of the term rather than just factoring out a constant from the derivative operator. This is an invalid application of the rule.gpt-oss:20b: fail (error) 2026-10-04 — Step 3 applies two transformations at once: it rewrites 24*x as 2*12*x and then applies the constant‑multiple rule, violating the one‑rule‑per‑step rule. The correct single step would be 24*Derivative(x,x).
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 2026-10-04 with SymPy 1.14.0.