Derivative of \( \displaystyle e^{\operatorname{atan}{\left(x \right)}} \)
Problem 2.548 · hard
Differentiate \( \displaystyle f(x) = e^{\operatorname{atan}{\left(x \right)}} \).
- \[ \frac{d}{d x} e^{\operatorname{atan}{\left(x \right)}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = e^{\operatorname{atan}{\left(x \right)}} \frac{d}{d x} \operatorname{atan}{\left(x \right)} \]chainApply the chain rule.✓ Proved
- \[ = \frac{e^{\operatorname{atan}{\left(x \right)}}}{x^{2} + 1} \]derivative algebraDifferentiate the inner function atan(x). Rearrange the terms for clarity.✓ Proved
Answer \( \frac{e^{\operatorname{atan}{\left(x \right)}}}{x^{2} + 1} \)
✓ 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 undefined where x**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x**2 + 1 = 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: pass — The solution correctly applies the chain rule and standard derivative formulas. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule and standard derivative formulas. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.