Derivative of \( \displaystyle 125 x^{3} - 75 x^{2} + 15 x \)
Problem 2.2050 · medium
Differentiate \( \displaystyle f(x) = 125 x^{3} - 75 x^{2} + 15 x \).
- \[ \frac{d}{d x} \left(125 x^{3} - 75 x^{2} + 15 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 15 x - \frac{d}{d x} 75 x^{2} + \frac{d}{d x} 125 x^{3} \]sumApply the sum rule.✓ Proved
- \[ = 15 \frac{d}{d x} x - 75 \frac{d}{d x} x^{2} + 125 \frac{d}{d x} x^{3} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 375 x^{2} - 150 x + 15 \]power algebra simplifyApply the power rule to each term. Simplify the exponents and coefficients. Simplify the expression.✓ Proved
Answer \( 15 \left(5 x - 1\right)^{2} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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: 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 the factored form in the stated answer matches the expanded result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the sum, constant-multiple, and power rules in separate steps. The final simplification is algebraically correct, and the factored form in the stated answer matches the expanded result.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the sum, constant-multiple, and power rules in sequence. The final answer is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.