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Derivative of \( \displaystyle 5 \operatorname{atan}{\left(x - 3 \right)} \)

Problem 2.595 · medium Video

Differentiate \( \displaystyle f(x) = 5 \operatorname{atan}{\left(x - 3 \right)} \).

\[ \frac{d}{d x} 5 \operatorname{atan}{\left(x - 3 \right)} \]
constant-multipleApply the constant multiple rule.✓ Proved
\[ = 5 \frac{d}{d x} \operatorname{atan}{\left(x - 3 \right)} \]
chainApply the chain rule to the arctangent function.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(x - 3\right)}{\left(x - 3\right)^{2} + 1} \]
derivativeDifferentiate the inner function x - 3.✓ Proved
\[ = \frac{5}{\left(x - 3\right)^{2} + 1} \]
algebra algebraSimplify the expression. Simplify the fraction.✓ Proved
\[ = \frac{5}{x^{2} - 6 x + 10} \]
algebra simplifyExpand the squared term. Combine like terms.✓ Proved

Answer \( \frac{5}{\left(x - 3\right)^{2} + 1} \)

Top: the graph of f = 5atan(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5/((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
6	✓ 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
7	✓ 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 (error) — Step 2 incorrectly labels the application of the chain rule; it should be a derivative of atan, not a chain rule. Step 3 combines two rules (derivative of atan and derivative of the inner function) in a single step, violating the one‑rule‑per‑step requirement.
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-10: pass — Every line is equivalent to the previous one, SymPy proved each step, and the final line matches the stated answer 5/((x-3)^2+1). The rule names used (constant-multiple, chain, derivative, algebra, simplify) are legitimate for the steps they describe.

gpt-oss:20b: dismiss — Differentiating atan(x-3) is a composition, so "chain" legitimately names the step, and writing 1/(1+(x-3)^2) * Derivative(x-3, x) is a single chain-rule application, not two independent rules.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — Every line is equivalent to the previous one, SymPy proved each step, and the final line matches the stated answer 5/((x-3)^2+1). The rule names used (constant-multiple, chain, derivative, algebra, simplify) are legitimate for the steps they describe.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 2 incorrectly labels the application of the chain rule; it should be a derivative of atan, not a chain rule. Step 3 combines two rules (derivative of atan and derivative of the inner function) in a single step, violating the one‑rule‑per‑step requirement.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: fail (error) 2026-09-21 — Step 2 applies the chain rule but does not actually perform the differentiation of the arctangent; step 3 then applies two rules at once (derivative of atan and derivative of the inner function). Each step must change only one rule, so step 3 should be split into two separate steps.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-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.