Derivative of \( \displaystyle x \left(5 x + 2\right) \)
Problem 2.85 · easy
Differentiate \( \displaystyle f(x) = x \left(5 x + 2\right) \).
- \[ \frac{d}{d x} x \left(5 x + 2\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 2 x + \frac{d}{d x} 5 x^{2} \]sum algebraApply the sum rule. Distribute x into the parentheses.✓ Proved
- \[ = 2 \frac{d}{d x} x + 5 \frac{d}{d x} x^{2} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 10 x + 2 \]power algebraApply the power rule to each term. Multiply the terms.✓ Proved
Answer \( 10 x + 2 \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 2 incorrectly applies the sum rule to a product; the derivative of x*(5*x+2) should use the product rule, not split into Derivative(x*(5*x)) + Derivative(2*x).qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 2 incorrectly applies the sum rule to the product x*(5*x+2); a product rule is required. The subsequent steps are based on this incorrect decomposition.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the sum rule, algebraic simplification, constant multiple rule, and power rule in distinct steps. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 2 incorrectly applies the sum rule to a product; the derivative of x*(5*x + 2) should use the product rule, not be split into two separate derivatives. Subsequent steps propagate this error.gpt-oss:20b: fail 2026-09-17 — Step 2 applies both distribution (x*(5*x+2)=x*(5*x)+2*x) and the sum rule in one line, yet it is labeled only as "sum". This mixes two rules and misleads the student.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.