Derivative of \( \displaystyle 6 x^{2} + 8 x \)
Problem 2.2003 · medium
Differentiate \( \displaystyle f(x) = 6 x^{2} + 8 x \).
- \[ \frac{d}{d x} \left(6 x^{2} + 8 x\right) \]sumStart with the derivative of the entire function.✓ Proved
- \[ = \frac{d}{d x} 8 x + \frac{d}{d x} 6 x^{2} \]sumApply the sum rule.✓ Proved
- \[ = 8 \frac{d}{d x} x + 6 \frac{d}{d x} x^{2} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 12 x + 8 \]power algebraApply the power rule to each term. Multiply the constants.✓ Proved
Answer \( 12 x + 8 \)
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: fail (error) — Step 1 incorrectly applies the sum rule to the entire function; the derivative of a sum should be written as the sum of the derivatives, not as a single derivative of the whole expression. The correct first step would be "Derivative(6*x**2, x) + Derivative(8*x, x)".qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — Step 1 incorrectly applies the sum rule to the entire function; the derivative of a sum should be written as the sum of the derivatives, not as a single derivative of the whole expression. The correct first step would be "Derivative(6*x**2, x) + Derivative(8*x, x)".qwen3.6:27b-mlx: pass 2026-10-10gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.