Derivative of \( \displaystyle e^{x^{2}} \)
Problem 2.1054 · hard
Differentiate \( \displaystyle f(x) = e^{x^{2}} \).
- \[ \frac{d}{d x} e^{x^{2}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = e^{x^{2}} \frac{d}{d x} x^{2} \]chainApply the chain rule.✓ Proved
- \[ = 2 x e^{x^{2}} \]derivative algebraDifferentiate the inner function x**2. Rearrange the terms for the final answer.✓ Proved
Answer \( 2 x e^{x^{2}} \)
✓ 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 |
| 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 (style) — Step 1 incorrectly labels the application of the chain rule as "derivative". The derivative of exp(x**2) should be obtained by the chain rule, not the basic derivative rule.qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and power rule in separate steps, adhering to the one-change-per-step constraint. The labels and notes are accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the chain rule and power rule in separate steps, adhering to the one-change-per-step constraint. The labels and notes are accurate.gpt-oss:20b: fail (style) 2026-09-28 — Step 1 incorrectly labels the application of the chain rule as "derivative". The derivative of exp(x**2) should be obtained by the chain rule, not the basic derivative rule.qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.