Derivative of \( \displaystyle 25 x^{3} + 15 x^{2} + 3 x \)
Problem 2.1492 · medium
Differentiate \( \displaystyle f(x) = 25 x^{3} + 15 x^{2} + 3 x \).
- \[ \frac{d}{d x} \left(25 x^{3} + 15 x^{2} + 3 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 3 x + \frac{d}{d x} 15 x^{2} + \frac{d}{d x} 25 x^{3} \]sumApply the sum rule.✓ Proved
- \[ = 3 \frac{d}{d x} x + 15 \frac{d}{d x} x^{2} + 25 \frac{d}{d x} x^{3} \]constant-multipleApply the constant multiple rule to each term.✓ Proved
- \[ = 75 x^{2} + 30 x + 3 \]power algebraApply the power rule to each term. Simplify the coefficients.✓ Proved
Answer \( 3 \left(5 x + 1\right)^{2} \)
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the sum, constant multiple, and power rules in separate steps. The final answer is algebraically equivalent to the derivative of the given function.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the sum, constant multiple, and power rules in separate steps. The final answer is algebraically equivalent to the derivative of the given function.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the sum, constant-multiple, and power rules in separate steps. The final algebraic simplification is correct, and the stated answer is algebraically equivalent to the result.gpt-oss:20b: fail (error) 2026-10-04 — The final simplified derivative 75*x**2 + 30*x + 3 is correct, but the stated answer 3*(5*x + 1)**2 is algebraically equivalent to 75*x**2 + 30*x + 3. However, the solution never shows the factorization step, so the final answer does not match the stated answer. This inconsistency is a defect.
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.