Derivative of \( \displaystyle \frac{\operatorname{acos}{\left(2 x \right)}}{2 x} \)
Problem 2.816 · hard
Differentiate \( \displaystyle f(x) = \frac{\operatorname{acos}{\left(2 x \right)}}{2 x} \).
- \[ \frac{d}{d x} \frac{\operatorname{acos}{\left(2 x \right)}}{2 x} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \operatorname{acos}{\left(2 x \right)} \frac{d}{d x} \frac{1}{2 x} + \frac{\frac{d}{d x} \operatorname{acos}{\left(2 x \right)}}{2 x} \]quotientApply the quotient rule.✓ Proved
- \[ = \operatorname{acos}{\left(2 x \right)} \frac{d}{d x} \frac{1}{2 x} - \frac{\frac{d}{d x} 2 x}{2 x \sqrt{1 - 4 x^{2}}} \]chainApply the chain rule to the numerator.✓ Proved
- \[ = \operatorname{acos}{\left(2 x \right)} \frac{d}{d x} \frac{1}{2 x} - \frac{1}{x \sqrt{1 - 4 x^{2}}} \]derivative algebra rewriteDifferentiate the inner function 2*x. Simplify the fraction. Rewrite 1/(2*x) as a power.✓ Proved
- \[ = - \frac{1}{x \sqrt{1 - 4 x^{2}}} - \frac{\operatorname{acos}{\left(2 x \right)} \frac{d}{d x} x}{2 x^{2}} \]powerApply the power rule.✓ Proved
- \[ = - \frac{1}{x \sqrt{1 - 4 x^{2}}} - \frac{\operatorname{acos}{\left(2 x \right)}}{2 x^{2}} \]derivative algebra simplifyDifferentiate x. Simplify the expression. Final simplification.✓ Proved
Answer \( - \frac{1}{x \sqrt{1 - 4 x^{2}}} - \frac{\operatorname{acos}{\left(2 x \right)}}{2 x^{2}} \)
✓ 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 undefined where 1 - 4*x**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where 1 - 4*x**2 = 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where 1 - 4*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 1 - 4*x**2 = 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where 1 - 4*x**2 = 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where 1 - 4*x**2 = 0 undefined where x = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where 1 - 4*x**2 = 0 undefined where x = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 acos is real only on [-1, 1] undefined where 1 - 4*x**2 = 0 undefined where x = 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 1 - 4*x**2 = 0 undefined where x = 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 2 incorrectly labels the operation as a quotient rule; the expression is actually differentiated using the product rule on acos(2*x) and 1/(2*x).qwen3.6:27b-mlx: pass — The solution correctly applies the quotient rule, chain rule, and power rule in separate steps. Each step modifies only one part of the expression as required, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the quotient rule, chain rule, and power rule in separate steps. Each step modifies only one part of the expression as required, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (style) 2026-09-26 — Step 2 incorrectly labels the operation as a quotient rule; the expression is actually differentiated using the product rule on acos(2*x) and 1/(2*x).qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the quotient rule, chain rule, and power rule in separate steps. Each step modifies only one part of the expression as required, and the labels 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.