Derivative of \( \displaystyle e^{\operatorname{atan}{\left(x - 3 \right)}} \)
Problem 2.377 · hard
Differentiate \( \displaystyle f(x) = e^{\operatorname{atan}{\left(x - 3 \right)}} \).
- \[ \frac{d}{d x} e^{\operatorname{atan}{\left(x - 3 \right)}} \]Start with the derivative of the function.✓ Proved
- \[ = e^{\operatorname{atan}{\left(x - 3 \right)}} \frac{d}{d x} \operatorname{atan}{\left(x - 3 \right)} \]chainApply the chain rule.✓ Proved
- \[ = \frac{e^{\operatorname{atan}{\left(x - 3 \right)}}}{\left(x - 3\right)^{2} + 1} \]derivative algebraDifferentiate the inner function atan(x - 3). Rewrite the expression as a single fraction.✓ Proved
- \[ = \frac{e^{\operatorname{atan}{\left(x - 3 \right)}}}{x^{2} - 6 x + 10} \]algebra simplifyExpand the denominator. Simplify the denominator.✓ Proved
Answer \( \frac{e^{\operatorname{atan}{\left(x - 3 \right)}}}{\left(x - 3\right)^{2} + 1} \)
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 undefined where (x - 3)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 0 undefined where x**2 - 6*x + 10 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 - 6*x + 10 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (x - 3)**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: fail (style) — Step 1 is labeled "unlabelled" but applies the derivative rule; it should be marked as "derivative". No other algebraic or rule‑application defects are present.qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and derivative rules, and the subsequent algebraic simplifications are valid and properly labeled.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule and derivative rules, and the subsequent algebraic simplifications are valid and properly labeled.gpt-oss:20b: fail (style) 2026-09-21 — Step 1 is labeled "unlabelled" but applies the derivative rule; it should be marked as "derivative". No other algebraic or rule‑application defects are present.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule and derivative rules for exponential and arctangent functions. All steps are algebraically sound and properly labeled.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 1 is unlabeled; it should be labeled "derivative". The step applies a single rule (taking the derivative of the whole expression) but the label is missing, which is a defect.
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.