Derivative of \( \displaystyle \frac{\operatorname{acos}{\left(x \right)}}{x} \)
Problem 2.385 · hard
Differentiate \( \displaystyle f(x) = \frac{\operatorname{acos}{\left(x \right)}}{x} \).
- \[ \frac{d}{d x} \frac{\operatorname{acos}{\left(x \right)}}{x} \]derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product.✓ Proved
- \[ = \operatorname{acos}{\left(x \right)} \frac{d}{d x} \frac{1}{x} + \frac{\frac{d}{d x} \operatorname{acos}{\left(x \right)}}{x} \]productApply the product rule.✓ Proved
- \[ = \operatorname{acos}{\left(x \right)} \frac{d}{d x} \frac{1}{x} - \frac{1}{x \sqrt{1 - x^{2}}} \]derivativeDifferentiate the first term.✓ Proved
- \[ = - \frac{1}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2}} \]derivative algebraDifferentiate the second term. Simplify the expression.✓ Proved
- \[ = \frac{- x - \sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}{x^{2} \sqrt{1 - x^{2}}} \]algebraCombine into a single fraction.✓ Proved
Answer \( - \frac{1}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2}} \)
✓ 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 acos is real only on [-1, 1] undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where x = 0 undefined where 1 - x**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where 1 - x**2 = 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where x = 0 undefined where 1 - x**2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where x = 0 undefined where 1 - x**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where x = 0 undefined where 1 - x**2 = 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 product rule and standard derivatives, with each step changing only one aspect of the expression. The final step combines terms into a single fraction, which is a valid algebraic simplification, even though the stated answer in the prompt was the separated form; the steps are logically sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule and standard derivatives, with each step changing only one aspect of the expression. The final step combines terms into a single fraction, which is a valid algebraic simplification, even though the stated answer in the prompt was the separated form; the steps are logically sound.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule and standard derivatives. Each step isolates a single operation, and the labels accurately reflect the rules applied.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule and standard derivative formulas. Each step isolates a single operation, and the labels accurately reflect the rules applied.gpt-oss:20b: pass 2026-09-20
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.