Derivative of \( \displaystyle \ln{\left(\ln{\left(2 x \right)}^{\frac{3}{2}} \right)} \)
Problem 2.1666 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(2 x \right)}^{\frac{3}{2}} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\ln{\left(2 x \right)}^{\frac{3}{2}} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{3 \ln{\left(\ln{\left(2 x \right)} \right)}}{2} \]algebraUse the logarithm power rule: log(a^b) = b*log(a).✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\ln{\left(2 x \right)} \right)}}{2} \]constant-multiplePull the constant out of the derivative.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(2 x \right)}}{2 \ln{\left(2 x \right)}} \]chainApply the chain rule to the nested logarithm.✓ Proved
- \[ = \frac{3 \frac{d}{d x} 2 x}{4 x \ln{\left(2 x \right)}} \]chainApply the chain rule to the inner logarithm.✓ Proved
- \[ = \frac{3}{2 x \ln{\left(2 x \right)}} \]derivative algebra simplifyDifferentiate the innermost function 2*x. Simplify the expression by canceling the 2s. Final simplification of the expression.✓ Proved
Answer \( \frac{3}{2 x \ln{\left(2 x \right)}} \)
Mind the domain. The answer is also defined on (0, 1/2), 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 undefined where log(2*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(2*x) = 0 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 log(2*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(2*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(2*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(2*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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the logarithm power rule, constant multiple rule, and chain rule in separate steps. The algebraic simplifications are correct and properly labeled.
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.