Derivative of \( \displaystyle 2 \ln{\left(\ln{\left(x - 3 \right)} \right)} \)
Problem 2.225 · medium
Differentiate \( \displaystyle f(x) = 2 \ln{\left(\ln{\left(x - 3 \right)} \right)} \).
- \[ \frac{d}{d x} 2 \ln{\left(\ln{\left(x - 3 \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 2 \frac{d}{d x} \ln{\left(\ln{\left(x - 3 \right)} \right)} \]constant-multiplePull the constant factor out of the derivative.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \ln{\left(x - 3 \right)}}{\ln{\left(x - 3 \right)}} \]chainApply the chain rule to the outer logarithm.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \left(x - 3\right)}{\left(x - 3\right) \ln{\left(x - 3 \right)}} \]chainApply the chain rule to the inner logarithm.✓ Proved
- \[ = \frac{2}{\left(x - 3\right) \ln{\left(x - 3 \right)}} \]derivative algebraDifferentiate the innermost term. Simplify the product of the terms.✓ Proved
Answer \( \frac{2}{\left(x - 3\right) \log{\left(x - 3 \right)}} \)
Mind the domain. The answer is also defined on (3, 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 undefined where log(x - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(x - 3) = 0 undefined where x - 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(x - 3) = 0 undefined where x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(x - 3) = 0 undefined where x - 3 = 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(x - 3) = 0 undefined where x - 3 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies the constant multiple rule and the chain rule in separate steps. Each step changes only one aspect of the expression, and the labels accurately describe the operations performed.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — The solution ignores the domain restriction for f(x)=2·log(log(x−3)). Since log(x−3) must be positive, we need x>4; the derivative is only valid on that interval. This omission is a domain/branch‑cut error.deepseek-r1:70b: pass 2026-09-17
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.