Derivative of \( \displaystyle \ln{\left(\ln{\left(3 x + 1 \right)} \right)} \)
Problem 2.1296 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(3 x + 1 \right)} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\ln{\left(3 x + 1 \right)} \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(3 x + 1 \right)}}{\ln{\left(3 x + 1 \right)}} \]chainApply the chain rule for the outer logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(3 x + 1\right)}{\left(3 x + 1\right) \ln{\left(3 x + 1 \right)}} \]chainApply the chain rule for the inner logarithm.✓ Proved
- \[ = \frac{3}{\left(3 x + 1\right) \ln{\left(3 x + 1 \right)}} \]derivative algebraDifferentiate the linear term 3*x + 1. Simplify the expression by multiplying the terms.✓ Proved
Answer \( \frac{3}{\left(3 x + 1\right) \ln{\left(3 x + 1 \right)}} \)
Mind the domain. The answer is also defined on (-1/3, 0), 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 undefined where log(3*x + 1) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x + 1) = 0 undefined where 3*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x + 1) = 0 undefined where 3*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x + 1) = 0 undefined where 3*x + 1 = 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(3*x + 1) = 0 undefined where 3*x + 1 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — Step 1 lacks a rule label; every step must specify the rule applied.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (style) 2026-10-03 — Step 1 lacks a rule label; every step must specify the rule applied.qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies the chain rule in distinct steps and uses valid labels from the fixed vocabulary. Each step changes only one aspect of the expression.gpt-oss:20b: fail (style) 2026-09-30 — Step 1 lacks a proper rule label; it should be marked as "derivative" rather than "unlabelled".
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 2026-10-03 with SymPy 1.14.0.