Derivative of \( \displaystyle x^{2} + 4 x \)
Problem 2.1581 · medium
Differentiate \( \displaystyle f(x) = x^{2} + 4 x \).
- \[ \frac{d}{d x} \left(x^{2} + 4 x\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 4 x + \frac{d}{d x} x^{2} \]sumApply the sum rule.✓ Proved
- \[ = 4 \frac{d}{d x} x + \frac{d}{d x} x^{2} \]constant-multipleApply the constant multiple rule to the second term.✓ Proved
- \[ = 2 x + 4 \]power simplifyDifferentiate the power term and the identity term. Simplify the final expression.✓ Proved
Answer \( 2 x + 4 \)
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 |
| 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: fail (error) — Step 4 applies two differentiation rules simultaneously (power rule for x**2 and derivative of identity for x), violating the one-rule-per-step constraint. It should be split into two steps.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 applies two differentiation rules simultaneously (power rule for x**2 and derivative of identity for x), violating the one-rule-per-step constraint. It should be split into two steps.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 applies two differentiation rules simultaneously (power rule for x^2 and derivative of identity for x), violating the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-10-05 — Step 1 applies no rule; it simply replaces the function with its derivative without justification. The correct first step should use the sum rule to split the derivative into the sum of the derivatives of the terms.
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 2026-10-05 with SymPy 1.14.0.