Derivative of \( \displaystyle \frac{\sqrt{x - 1}}{x} \)
Problem 2.564 · hard
Differentiate \( \displaystyle f(x) = \frac{\sqrt{x - 1}}{x} \).
- \[ \frac{d}{d x} \frac{\sqrt{x - 1}}{x} \]algebraStart with the derivative of the function. Rewrite the quotient as a product.✓ Proved
- \[ = \sqrt{x - 1} \frac{d}{d x} \frac{1}{x} + \frac{\frac{d}{d x} \sqrt{x - 1}}{x} \]product rewriteApply the product rule. Rewrite the square root as a fractional power.✓ Proved
- \[ = \sqrt{x - 1} \frac{d}{d x} \frac{1}{x} + \frac{1}{2 x \sqrt{x - 1}} \]powerApply the power rule to the first term.✓ Proved
- \[ = \frac{1}{2 x \sqrt{x - 1}} - \frac{\sqrt{x - 1}}{x^{2}} \]derivative algebraDifferentiate the second term. Simplify the terms.✓ Proved
- \[ = \frac{2 - x}{2 x^{2} \sqrt{x - 1}} \]algebra simplifyFind a common denominator. Simplify the numerator.✓ Proved
Answer \( \frac{2 - x}{2 x^{2} \sqrt{x - 1}} \)
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 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x - 1 = 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 (error) — Step 7 incorrectly combines the two terms; the common denominator and resulting numerator are wrong, leading to an incorrect final derivative.qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, power rule, and algebraic simplification steps. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, power rule, and algebraic simplification steps. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-21 — Step 7 incorrectly combines the two terms; the common denominator and resulting numerator are wrong, leading to an incorrect final derivative.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule and subsequent differentiation steps. Each step isolates a single operation, and the labels accurately reflect the rules applied.gpt-oss:20b: pass 2026-09-21
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.