Derivative of \( \displaystyle 75 x^{3} + 45 x^{2} + 9 x \)
Problem 2.2391 · medium Video
Differentiate \( \displaystyle f(x) = 75 x^{3} + 45 x^{2} + 9 x \).
- \[ \frac{d}{d x} \left(75 x^{3} + 45 x^{2} + 9 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 9 x + \frac{d}{d x} 45 x^{2} + \frac{d}{d x} 75 x^{3} \]sumApply the sum rule.✓ Proved
- \[ = 9 \frac{d}{d x} x + 45 \frac{d}{d x} x^{2} + 75 \frac{d}{d x} x^{3} \]constant-multipleApply the constant multiple rule to each term.✓ Proved
- \[ = 225 x^{2} + 90 x + 9 \]power simplifyApply the power rule to each term. Simplify the coefficients.✓ Proved
Answer \( 9 \left(5 x + 1\right)^{2} \)
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. How we verify
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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: fail (error) 2026-10-16 — The final stated answer 9*(5*x + 1)**2 does not equal the correct derivative 225*x**2 + 90*x + 9 derived in the steps. The solution’s answer is mathematically incorrect.qwen3.6:27b-mlx: pass 2026-10-15gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.