Derivative of \( \displaystyle e^{4 x^{2} + 1} \)
Problem 2.333 · hard
Differentiate \( \displaystyle f(x) = e^{4 x^{2} + 1} \).
- \[ \frac{d}{d x} e^{4 x^{2} + 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = e^{4 x^{2} + 1} \frac{d}{d x} \left(4 x^{2} + 1\right) \]chainApply the chain rule.✓ Proved
- \[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} 4 x^{2}\right) e^{4 x^{2} + 1} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = 8 x e^{4 x^{2} + 1} \]power constant simplifyDifferentiate the power term. The derivative of the constant 1 is 0. Simplify the final expression.✓ Proved
Answer \( 8 x e^{4 x^{2} + 1} \)
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 (13)
qwen3.6:27b-mlx: pass 2026-09-20gpt-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: fail (error) 2026-09-19 — Step 4 applies two rules at once (power and constant‑multiple) but is labeled only as "power". Each step must change only one thing and name the rule correctly.qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 4 is labeled 'power' but applies the power rule to 4*x**2 while leaving the derivative of 1 untouched; strictly, the power rule applies to the term 4*x**2, but the step also implicitly handles the constant term's derivative (or rather, the previous step split them, so this step should only address the power term, but the label 'power' is acceptable for 4*x**2 -> 8*x. However, Step 5 is labeled 'constant' but performs the simplification 8*x + 0 -> 8*x, which is algebraic simplification, not the application of the constant rule (which was already applied in step 3/4 logic). Actually, looking closer: Step 3 splits the derivative into a sum. Step 4 computes Derivative(4*x**2, x) as 8*x using 'power'. It leaves Derivative(1, x) as is? No, Step 4 shows (8*x + 0). This implies Step 4 computed BOTH derivatives: 4*x**2 -> 8*x (power) and 1 -> 0 (constant). Applying two rules (power and constant) in one step is a defect. Step 5 then simplifies 8*x + 0 to 8*x, labeled 'constant', which is incorrect labeling for simplification.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 4 applies both the constant‑multiple rule (derivative of 4·x²) and the power rule in one go, violating the one‑rule‑per‑step rule. Step 5 then drops the zero term, which is a simplification, yet it is labeled as a constant rule. Both steps combine multiple transformations and mislabel the applied rule.
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.