Derivative of \( \displaystyle \frac{2 \operatorname{atan}{\left(5 x \right)}}{5} \)
Problem 2.670 · medium
Differentiate \( \displaystyle f(x) = \frac{2 \operatorname{atan}{\left(5 x \right)}}{5} \).
- \[ \frac{d}{d x} \frac{2 \operatorname{atan}{\left(5 x \right)}}{5} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \operatorname{atan}{\left(5 x \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{2 \frac{d}{d x} 5 x}{5 \left(25 x^{2} + 1\right)} \]chainApply the chain rule to the arctangent function.✓ Proved
- \[ = \frac{2}{25 x^{2} + 1} \]power algebraDifferentiate the inner function 5*x. Simplify the expression by canceling the 5.✓ Proved
Answer \( \frac{2}{25 x^{2} + 1} \)
✓ 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.
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 (style) — Step 4 applies two transformations—simplifying (5*x)**2 to 25*x**2 and multiplying by 5—yet it is labeled only as "power". The label should reflect the simplification or the derivative of the inner function, not both at once.qwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'power', but the operation performed is differentiating a linear term (5*x), which is a basic derivative rule, not the power rule. The label should be 'derivative'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 4 is labeled 'power', but the operation performed is differentiating a linear term (5*x), which is a basic derivative rule, not the power rule. The label should be 'derivative'.gpt-oss:20b: fail (style) 2026-09-21 — Step 4 applies two transformations—simplifying (5*x)**2 to 25*x**2 and multiplying by 5—yet it is labeled only as "power". The label should reflect the simplification or the derivative of the inner function, not both at once.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule, and power rule in separate steps. The labels accurately reflect the operations performed, and the algebraic simplification is correct.gpt-oss:20b: fail (style) 2026-09-21 — Step 4 incorrectly labels the differentiation of 5*x as "power"; it should be "constant-multiple" or simply "derivative". The algebraic simplification in step 5 is fine.
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.