Derivative of \( \displaystyle 25 x^{3} - 15 x^{2} + 3 x \)
Problem 2.2174 · 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 simplification is algebraically correct and matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies the sum, constant-multiple, and power rules in separate steps. The final simplification is algebraically correct and matches the stated answer.gpt-oss:20b: fail (error) 2026-10-13 — The final simplified derivative 75*x**2 - 30*x + 3 does not equal the stated answer 3*(5*x - 1)**2. The solution incorrectly claims the two expressions are equivalent, which is mathematically false.qwen3.6:27b-mlx: pass 2026-10-13
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 2026-10-13 with SymPy 1.14.0.