Home›Calculus 1›Derivatives of inverse trig functions›Problem 2.377

Derivative of \( \displaystyle e^{\operatorname{atan}{\left(x - 3 \right)}} \)

Problem 2.377 · hard Video

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 \( \frac{e^{\operatorname{atan}{\left(x - 3 \right)}}}{\left(x - 3\right)^{2} + 1} \)

Top: the graph of f = e^(atan(x - 3)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = e^(atan(x - 3))/((x - 3)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. How we verify

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 x2 - 6*x + 10 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 - 6x + 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.

Senior review claude-sonnet-5-5, 2026-10-08: pass — The chain rule step, the derivative of atan(x-3), and the algebra and simplify steps are all correct, single-change, and properly labelled. The final line matches the stated answer, since 1+(x-3)^2 = x^2-6x+10.

gpt-oss:20b: dismiss — Line 1 simply restates the problem and SymPy proved it matches the task, so it needs no rule label; no real defect exists.
(panel model that both passed and objected): dismiss — No specific defect was stated, and every line was proved equal by SymPy with correct rule labels.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-08 — The chain rule step, the derivative of atan(x-3), and the algebra and simplify steps are all correct, single-change, and properly labelled. The final line matches the stated answer, since 1+(x-3)^2 = x^2-6x+10.
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 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-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.