Derivative of \( \displaystyle 18 x^{3} - 54 x^{2} + 54 x \)
Problem 2.2197 · medium Beautiful
Differentiate \( \displaystyle f(x) = 18 x^{3} - 54 x^{2} + 54 x \).
- \[ \frac{d}{d x} \left(18 x^{3} - 54 x^{2} + 54 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 54 x - \frac{d}{d x} 54 x^{2} + \frac{d}{d x} 18 x^{3} \]sumApply the sum rule.✓ Proved
- \[ = 54 \frac{d}{d x} x - 54 \frac{d}{d x} x^{2} + 18 \frac{d}{d x} x^{3} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 54 x^{2} - 108 x + 54 \]power algebra simplifyApply the power rule to each term. Multiply the constants. Factor out the greatest common factor.✓ Proved
Answer \( 54 \left(x - 1\right)^{2} \)
✓ 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 differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies the sum rule, constant multiple rule, and power rule in distinct steps. The final simplification is algebraically correct and properly labeled.
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.