Derivative of \( \displaystyle \operatorname{atan}{\left(\frac{1}{x} \right)} \)
Problem 2.817 · hard
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\frac{1}{x} \right)} \).
- \[ \frac{d}{d x} \operatorname{atan}{\left(\frac{1}{x} \right)} \]derivative algebraStart with the derivative of the function. Rewrite 1/x as x**-1.✓ Proved
- \[ = \frac{\frac{d}{d x} \frac{1}{x}}{1 + \frac{1}{x^{2}}} \]inverse-trig powerApply the derivative rule for arctan(u). Simplify the squared term.✓ Proved
- \[ = - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)} \]power algebra algebraDifferentiate x**-1 using the power rule. Combine the fraction and the derivative. Rewrite x**-2 as 1/x**2.✓ Proved
- \[ = - \frac{1}{x^{2} + 1} \]algebra simplifySimplify the complex fraction by multiplying by the reciprocal. Cancel the x**2 terms to get the final result.✓ Proved
Answer \( - \frac{1}{x^{2} + 1} \)
Mind the domain. The answer is also defined on {0}, where f(x) is not. Substituting there gives a number that is not a slope of f.
✓ 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 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 + x**(-2) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 + x**(-2) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 + x**(-2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 + x**(-2) = 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 + x**(-2) = 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 + x**(-2) = 0 undefined where x = 0 undefined where x**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and power rule in distinct steps, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule and power rule in distinct steps, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule and power rule, with each step changing only one aspect of the expression. The labels used are from the allowed vocabulary and accurately describe the operations performed.gpt-oss:20b: pass 2026-09-26
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.