Derivative of \( \displaystyle 32 x^{3} - 72 x^{2} + 54 x \)
Problem 2.2277 · medium Video
Differentiate \( \displaystyle f(x) = 32 x^{3} - 72 x^{2} + 54 x \).
- \[ \frac{d}{d x} \left(32 x^{3} - 72 x^{2} + 54 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 54 x - \frac{d}{d x} 72 x^{2} + \frac{d}{d x} 32 x^{3} \]sumApply the sum rule.✓ Proved
- \[ = 54 \frac{d}{d x} x - 72 \frac{d}{d x} x^{2} + 32 \frac{d}{d x} x^{3} \]constant-multipleApply the constant multiple rule to each term.✓ Proved
- \[ = 96 x^{2} - 144 x + 54 \]power algebraApply the power rule to each term. Simplify the coefficients and terms.✓ Proved
Answer \( 6 \left(4 x - 3\right)^{2} \)
✓ 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 |
| 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 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the sum, constant-multiple, and power rules in separate steps. The final simplification is algebraically correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.