Derivative of \( \displaystyle 10 x \left(x - 3\right) \)
Problem 2.96 · easy
Differentiate \( \displaystyle f(x) = 10 x \left(x - 3\right) \).
- \[ \frac{d}{d x} 10 x \left(x - 3\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(10 x^{2} - 30 x\right) \]algebraExpand the expression using the distributive property.✓ Proved
- \[ = - \frac{d}{d x} 30 x + \frac{d}{d x} 10 x^{2} \]sumApply the difference rule for derivatives.✓ Proved
- \[ = - 30 \frac{d}{d x} x + 10 \frac{d}{d x} x^{2} \]constant-multipleFactor out the constants from each term.✓ Proved
- \[ = 20 x - 30 \]power algebraApply the power rule to each term. Multiply the constants to simplify the expression.✓ Proved
Answer \( 20 x - 30 \)
✓ 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies the distributive property, linearity of the derivative, constant multiple rule, and power rule in separate steps. All labels are from the fixed vocabulary and accurately describe the transformations.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the distributive property, linearity of the derivative, constant multiple rule, and power rule in separate steps. All labels are from the fixed vocabulary and accurately describe the transformations.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 3 labels the rule as 'sum' while the note and operation actually apply the difference rule for derivatives, which is a mismatch.deepseek-r1:70b: pass 2026-09-17
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 2026-09-26 with SymPy 1.14.0.