Derivative of \( \displaystyle \frac{\operatorname{atan}{\left(5 x \right)}}{5} \)
Problem 2.615 · medium
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 \( \frac{1}{25 x^{2} + 1} \)
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 25*x**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 25*x**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 25*x**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 25*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: fail (error) — 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-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 2026-09-21gpt-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.