Derivative of \( \displaystyle - \ln{\left(\ln{\left(x + 2 \right)} \right)} \)
Problem 2.836 · hard
Differentiate \( \displaystyle f(x) = - \ln{\left(\ln{\left(x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\ln{\left(x + 2 \right)} \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\ln{\left(x + 2 \right)} \right)} \]constantPull out the constant factor -1.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(x + 2 \right)}}{\ln{\left(x + 2 \right)}} \]chainApply the chain rule for the outer logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(x + 2\right)}{\left(x + 2\right) \ln{\left(x + 2 \right)}} \]chainApply the chain rule for the inner logarithm.✓ Proved
- \[ = - \frac{1}{\left(x + 2\right) \ln{\left(x + 2 \right)}} \]derivative simplifyDifferentiate the innermost term. Simplify the expression.✓ Proved
Answer \( - \frac{1}{\left(x + 2\right) \ln{\left(x + 2 \right)}} \)
Mind the domain. The answer is also defined on (-2, -1), 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 + 2) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(x + 2) = 0 undefined where x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 undefined where log(x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 undefined where log(x + 2) = 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 + 2 = 0 undefined where log(x + 2) = 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 in successive steps, properly handles the constant factor, and simplifies the final expression. Each step changes only one aspect of the expression as required.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule in successive steps, properly handles the constant factor, and simplifies the final expression. Each step changes only one aspect of the expression as required.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule in distinct steps and uses valid labels from the fixed vocabulary. The algebraic simplification in the final step is correct.gpt-oss:20b: fail (style) 2026-09-26 — Step 1 lacks a rule label; it should be marked with the "derivative" rule to satisfy the labeling requirement.
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.