Derivative of \( \displaystyle \ln{\left(\sqrt{\ln{\left(4 x \right)}} \right)} \)
Problem 2.1689 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\sqrt{\ln{\left(4 x \right)}} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\sqrt{\ln{\left(4 x \right)}} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\ln{\left(4 x \right)}^{0.5} \right)} \]rewriteRewrite the square root using the exponential and logarithm rule.✓ Proved
- \[ = \frac{d}{d x} 0.5 \ln{\left(\ln{\left(4 x \right)} \right)} \]algebraApply the property of logarithms to move the constant outside.✓ Proved
- \[ = 0.5 \frac{d}{d x} \ln{\left(\ln{\left(4 x \right)} \right)} \]constant-multipleMove the constant multiplier outside the derivative.✓ Proved
- \[ = \frac{0.5 \frac{d}{d x} \ln{\left(4 x \right)}}{\ln{\left(4 x \right)}} \]chainApply the chain rule to the outer logarithm.✓ Proved
- \[ = \frac{0.125 \frac{d}{d x} 4 x}{x \ln{\left(4 x \right)}} \]chainApply the chain rule to the inner logarithm.✓ Proved
- \[ = \frac{0.5}{x \ln{\left(4 x \right)}} \]derivativeDifferentiate the innermost linear function.✓ Proved
- \[ = \frac{1}{2 x \ln{\left(4 x \right)}} \]algebra simplifySimplify the constants in the numerator and denominator. Simplify the final expression.✓ Proved
Answer \( \frac{1}{2 x \ln{\left(4 x \right)}} \)
Mind the domain. The answer is also defined on (0, 1/4), 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.
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 |
| 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 undefined where log(4*x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(4*x) = 0 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 log(4*x) = 0 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 log(4*x) = 0 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 log(4*x) = 0 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 log(4*x) = 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and logarithmic properties in distinct steps. The labels accurately reflect the operations performed, and the final simplification is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the chain rule and logarithmic properties in distinct steps. The labels accurately reflect the operations performed, and the final simplification is correct.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the chain rule and logarithmic properties in distinct steps. Each step modifies only one aspect of the expression, and the labels accurately reflect the operations performed.
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 2026-10-06 with SymPy 1.14.0.