Home›Calculus 1›Chain rule›Problem 2.1033

Derivative of \( \displaystyle \ln{\left(\frac{\sqrt{1 - x^{2}}}{x} \right)} \)

Problem 2.1033 · hard Beautiful

Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{1 - x^{2}}}{x} \right)} \).

\[ \frac{d}{d x} \ln{\left(\frac{\sqrt{1 - x^{2}}}{x} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(- \ln{\left(x \right)} + \ln{\left(\sqrt{1 - x^{2}} \right)}\right) \]
algebra algebraUse the property log(a/b) = log(a) - log(b). Rewrite the square root as a power.✓ Proved
\[ = \frac{d}{d x} \left(- \ln{\left(x \right)} + \frac{\ln{\left(1 - x^{2} \right)}}{2}\right) \]
constant-multiplePull out the constant factor 1/2.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(x \right)} + \frac{d}{d x} \frac{\ln{\left(1 - x^{2} \right)}}{2} \]
sumApply the sum rule for derivatives.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(x \right)} + \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} \]
constant-multipleDistribute the derivative operator.✓ Proved
\[ = \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \left(1 - x^{2}\right)} - \frac{1}{x} \]
logarithmicApply the derivative rule for the natural logarithm.✓ Proved
\[ = - \frac{x}{1 - x^{2}} - \frac{1}{x} \]
derivative algebraDifferentiate the inner function 1 - x**2. Simplify the expression by multiplying 1/2 and -2*x.✓ Proved
\[ = - \frac{1}{x \left(1 - x^{2}\right)} \]
algebra algebra simplifyFind a common denominator. Expand the numerator. Simplify the final expression.✓ Proved

Answer \( \frac{1}{x^{3} - x} \)

Mind the domain. The answer is also defined on (-oo, -1) and (-1, 0) and (1, oo), 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. 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 log is undefined for non-positive arguments undefined where x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments 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
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - x**2 = 0 undefined where x = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - x**2 = 0 undefined where x = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - x**2 = 0 undefined where x = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - 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 undefined where x**3 - x = 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the rules of differentiation and algebraic simplification. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-09-27

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 2026-09-28 with SymPy 1.14.0.