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

Problem 2.615 · medium Video

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

\[ \frac{d}{d x} \frac{\operatorname{atan}{\left(5 x \right)}}{5} \]
constant-multipleExtract the constant factor 1/5.✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(5 x \right)}}{5} \]
chainApply the chain rule to the inner function 5*x.✓ Proved
\[ = \frac{\frac{d}{d x} 5 x}{5 \left(25 x^{2} + 1\right)} \]
derivativeDifferentiate the inner function 5*x.✓ Proved
\[ = \frac{1}{25 x^{2} + 1} \]
algebra simplifyExpand the square in the denominator. Cancel the common factor of 5.✓ Proved

Answer \( \frac{1}{25 x^{2} + 1} \)

Top: the graph of f = atan(5x)/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 1/(25x² + 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 25x*2 + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 25x*2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 25x*2 + 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 25x*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 3 applies two rules at once (derivative of atan and derivative of the inner 5*x) and is incorrectly labeled as "derivative".
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-10: fail — The math is correct and SymPy proved every line. The rule labels and notes are misaligned with the steps: the chain-rule label sits on the constant-multiple step, and the evaluation of d(5x)=5 is combined with expanding the square under 'algebra'. Evaluating Derivative(5*x) should be folded into the chain step or given its own step.

gpt-oss:20b: uphold — The step labels don't match the steps: line 3 is labeled 'derivative' (note: differentiate 5*x) but actually applies the atan/chain rule and leaves Derivative(5*x) unevaluated, and the next step both evaluates d(5x)=5 and expands (5x)^2 under the single label 'algebra'. The reasoning is not exactly as the panel stated, but a real labeling defect exists.

Every verdict on record (7)

claude-sonnet-5-5: fail (style) 2026-10-10 — The math is correct and SymPy proved every line. The rule labels and notes are misaligned with the steps: the chain-rule label sits on the constant-multiple step, and the evaluation of d(5x)=5 is combined with expanding the square under 'algebra'. Evaluating Derivative(5*x) should be folded into the chain step or given its own step.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 3 applies two rules at once (derivative of atan and derivative of the inner 5*x) and is incorrectly labeled as "derivative".
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: fail (error) 2026-09-21 — Step 2 applies the chain rule in addition to the constant‑multiple extraction, violating the one‑rule‑per‑step rule. Step 3 simultaneously differentiates the outer atan and the inner 5*x, applying two rules at once. Both steps should be split into separate lines with appropriate labels.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: fail (style) 2026-09-21 — Step 2 is labeled "chain" but only a constant‑multiple rule was applied; the chain rule is applied in step 3. This mislabeling is a style 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.