Derivative of \( \displaystyle 75 x^{3} + 90 x^{2} + 36 x \)
Problem 2.2195 · medium
Differentiate \( \displaystyle f(x) = 75 x^{3} + 90 x^{2} + 36 x \).
- \[ \frac{d}{d x} \left(75 x^{3} + 90 x^{2} + 36 x\right) \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 36 x + \frac{d}{d x} 90 x^{2} + \frac{d}{d x} 75 x^{3} \]constant-multiplePull out the constants.✓ Proved
- \[ = 36 \frac{d}{d x} x + 90 \frac{d}{d x} x^{2} + 75 \frac{d}{d x} x^{3} \]powerApply the power rule to each term.✓ Proved
- \[ = 225 x^{2} + 180 x + 36 \]algebra simplifySimplify the coefficients and exponents. Final simplified expression.✓ Proved
Answer \( 9 \left(5 x + 2\right)^{2} \)
✓ 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 |
| 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: fail (style) — Step 2 applies the constant-multiple rule to all three terms simultaneously, violating the 'one thing per step' constraint. It should be split into three separate steps or labeled as applying the rule to a single term at a time.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (style) 2026-10-13 — Step 2 applies the constant-multiple rule to all three terms simultaneously, violating the 'one thing per step' constraint. It should be split into three separate steps or labeled as applying the rule to a single term at a time.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-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.