Derivative of \( \displaystyle 20 x^{3} + 60 x^{2} + 60 x \)
Problem 2.1651 · medium Beautiful
Differentiate \( \displaystyle f(x) = 20 x^{3} + 60 x^{2} + 60 x \).
- \[ \frac{d}{d x} \left(20 x^{3} + 60 x^{2} + 60 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 60 x + \frac{d}{d x} 60 x^{2} + \frac{d}{d x} 20 x^{3} \]sumApply the sum rule.✓ Proved
- \[ = 6 \frac{d}{d x} 10 x + 60 \frac{d}{d x} x^{2} + 20 \frac{d}{d x} x^{3} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 60 \frac{d}{d x} x + 60 \frac{d}{d x} x^{2} + 20 \frac{d}{d x} x^{3} \]algebraSimplify the constant terms.✓ Proved
- \[ = 60 x^{2} + 120 x + 60 \]power algebraApply the power rule to each term. Multiply the coefficients.✓ Proved
Answer \( 60 \left(x + 1\right)^{2} \)
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 |
| 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 60*x as 6*(10*x) and then pulls out the constant 6, which is not a single allowed rule. Moreover, the resulting expression 6*Derivative(10*x) is not algebraically equivalent to the intended 60*Derivative(x), so the step is mathematically incorrect.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: fail (error) 2026-10-06 — Step 3 applies two rules at once: it rewrites 60*x as 6*(10*x) and then pulls out the constant 6, which is not a single allowed rule. Moreover, the resulting expression 6*Derivative(10*x) is not algebraically equivalent to the intended 60*Derivative(x), so the step is mathematically incorrect.qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: fail (error) 2026-10-06 — Step 3 rewrites 60 as 6·10 and then differentiates 10·x, applying two rules at once and producing an incorrect intermediate expression. The correct constant‑multiple step should give 60·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 2026-10-06 with SymPy 1.14.0.