Derivative of \( \displaystyle \frac{\ln{\left(2 x \right)}}{4 x^{2}} \)
Problem 2.456 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(2 x \right)}}{4 x^{2}} \).
- \[ \frac{d}{d x} \frac{\ln{\left(2 x \right)}}{4 x^{2}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(2 x \right)}}{4 x^{2}} - \frac{\ln{\left(2 x \right)} \frac{d}{d x} 4 x^{2}}{16 x^{4}} \]quotientApply the quotient rule.✓ Proved
- \[ = - \frac{\ln{\left(2 x \right)}}{2 x^{3}} + \frac{1}{4 x^{3}} \]chain algebra algebra algebraDifferentiate the numerator using the chain rule. Simplify the numerator of the first term. Simplify the first term. Simplify the second term.✓ Proved
- \[ = \frac{1 - 2 \ln{\left(2 x \right)}}{4 x^{3}} \]algebraCombine the terms over a common denominator.✓ Proved
- \[ = \frac{- 2 \ln{\left(x \right)} - 2 \ln{\left(2 \right)} + 1}{4 x^{3}} \]algebra algebraExpand the logarithm. Distribute the constant.✓ Proved
- \[ = \frac{1 - 2 \ln{\left(2 x \right)}}{4 x^{3}} \]simplifyFinal simplified form.✓ Proved
Answer \( \frac{1 - 2 \log{\left(2 x \right)}}{4 x^{3}} \)
✓ 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 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 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 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 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the quotient rule and chain rule, followed by valid algebraic simplifications. The final steps expanding and recombining the logarithm are redundant but mathematically correct and do not violate the single-rule-per-step constraint.
Senior review claude-sonnet-5-5, 2026-10-09: fail — All lines are mathematically correct and the final answer matches. Line 4 does two things while its note mentions only one: it simplifies the first term and also expands (4*x**2)**2 to 16*x**4 in the second. Lines 8-10 expand log(2*x) and then undo it, which adds nothing. Line 8's 'algebra' label could be 'logarithmic', but either fits.
Every verdict on record (7)
claude-sonnet-5-5: fail (style) 2026-10-09 — All lines are mathematically correct and the final answer matches. Line 4 does two things while its note mentions only one: it simplifies the first term and also expands (4*x**2)**2 to 16*x**4 in the second. Lines 8-10 expand log(2*x) and then undo it, which adds nothing. Line 8's 'algebra' label could be 'logarithmic', but either fits.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the quotient rule and chain rule, followed by valid algebraic simplifications. The final steps expanding and recombining the logarithm are redundant but mathematically correct and do not violate the single-rule-per-step constraint.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'chain' but performs two distinct operations: differentiating the numerator (chain rule) and the denominator (power rule). The contract requires each step to change only one thing.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the quotient rule, chain rule, and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the rules applied.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.