Derivative of \( \displaystyle 27 x^{3} - 81 x^{2} + 81 x \)
Problem 2.2417 · medium Beautiful Video
Differentiate \( \displaystyle f(x) = 27 x^{3} - 81 x^{2} + 81 x \).
- \[ \frac{d}{d x} \left(27 x^{3} - 81 x^{2} + 81 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 81 x - \frac{d}{d x} 81 x^{2} + \frac{d}{d x} 27 x^{3} \]sumApply the sum rule.✓ Proved
- \[ = 81 \frac{d}{d x} x - 81 \frac{d}{d x} x^{2} + 27 \frac{d}{d x} x^{3} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 81 x^{2} - 162 x + 81 \]power algebra simplifyApply the power rule to each term. Multiply the coefficients. Factor out the common term.✓ Proved
Answer \( 81 \left(x - 1\right)^{2} \)
✓ 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. 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 |
| 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 clearly labeled.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the sum, constant-multiple, and power rules in separate steps. The final simplification is algebraically correct and clearly labeled.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the sum rule, constant multiple rule, and power rule in separate steps. The final simplification is algebraically correct and clearly labeled.gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.