Home›Calculus 1›Quotient rule›Problem 2.564

Derivative of \( \displaystyle \frac{\sqrt{x - 1}}{x} \)

Problem 2.564 · hard Video

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 \( \frac{2 - x}{2 x^{2} \sqrt{x - 1}} \)

Top: the graph of f = sqrt(x - 1)/x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 1/(2x·sqrt(x - 1)) - sqrt(x - 1)/x².

✓ 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. How we verify

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 = 0 undefined where x - 1 = 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 = 0 undefined where x - 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 0 undefined where x - 1 = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the product rule and power rule, with appropriate algebraic simplifications at each step. All labels are valid and correspond to the operations performed.

Senior review claude-sonnet-5-5, 2026-10-10: pass — Every line is equivalent to the previous one, and the labels are legitimate. The derivative (2 - x)/(2x^2 sqrt(x - 1)) is correct. The panel listed no specific objections, and none of the steps does two independent things.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — Every line is equivalent to the previous one, and the labels are legitimate. The derivative (2 - x)/(2x^2 sqrt(x - 1)) is correct. The panel listed no specific objections, and none of the steps does two independent things.
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule and power rule, with appropriate algebraic simplifications at each step. All labels are valid and correspond to the operations performed.
gpt-oss:20b: pass 2026-09-26
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.