Derivative of \( \displaystyle \ln{\left(\frac{\sqrt{4 x^{2} - 1}}{2 x} \right)} \)
Problem 2.532 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{4 x^{2} - 1}}{2 x} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\frac{\sqrt{4 x^{2} - 1}}{2 x} \right)} \]derivative rewrite rewriteRewrite the square root using a fractional exponent. Rewrite the logarithm as an exponential of a logarithm.✓ Proved
- \[ = \frac{d}{d x} \left(- \ln{\left(2 x \right)} + \ln{\left(\sqrt{4 x^{2} - 1} \right)}\right) \]simplifyUse logarithm properties to split the quotient.✓ Proved
- \[ = \frac{d}{d x} \left(- \ln{\left(2 x \right)} + 0.5 \ln{\left(4 x^{2} - 1 \right)}\right) \]algebraBring the exponent to the front.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(2 x \right)} + 0.5 \frac{d}{d x} \ln{\left(4 x^{2} - 1 \right)} \]constant-multiple✓ Proved
- \[ = 0.5 \frac{d}{d x} \ln{\left(4 x^{2} - 1 \right)} - \frac{\frac{d}{d x} 2 x}{2 x} \]chainApply the chain rule to the second term.✓ Proved
- \[ = \frac{0.5 \frac{d}{d x} \left(4 x^{2} - 1\right)}{4 x^{2} - 1} - \frac{1}{x} \]logarithmicDifferentiate the logarithm.✓ Proved
- \[ = \frac{4.0 x}{4 x^{2} - 1} - \frac{1}{x} \]derivativeDifferentiate the inner polynomial.✓ Proved
- \[ = \frac{4 x}{4 x^{2} - 1} - \frac{1}{x} \]algebraSimplify the coefficients.✓ Proved
- \[ = \frac{1}{x \left(4 x^{2} - 1\right)} \]algebra simplifyFind a common denominator. Simplify the numerator.✓ Proved
Answer \( \frac{1}{4 x^{3} - x} \)
Mind the domain. The answer is also defined on (-oo, -1/2) and (-1/2, 0) and (0, 1/2), where f(x) is not. Substituting there gives a number that is not a slope of f.
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 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 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 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 x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 undefined where 4*x**2 - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 1 = 0 undefined where x = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 1 = 0 undefined where x = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 1 = 0 undefined where x = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 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 4*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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint, and the labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is mathematically incorrect; it replaces log(u) with log(exp(log(u))), which is not an identity (it adds an unnecessary layer of composition and changes the expression structure without justification). Furthermore, Step 4 jumps from the incorrect Step 3 to a split logarithm, applying multiple rules (simplifying the exp/log composition and splitting the quotient) in one step, violating the 'one change per step' rule.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 is a redundant and confusing rewrite (log(exp(log(u))) = log(u)) that adds no value and obscures the direct application of logarithm properties in Step 4. Step 6 applies the linearity of the derivative (sum rule) to split the expression, but is labeled 'constant-multiple'; while constant-multiple is applied to the first term, the splitting itself is a sum rule operation, making the label incomplete or misleading for the whole step.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.