Home›Calculus 1›Quotient rule›Problem 2.836

Derivative of \( \displaystyle - \ln{\left(\ln{\left(x + 2 \right)} \right)} \)

Problem 2.836 · hard Video

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 \( - \frac{1}{\left(x + 2\right) \ln{\left(x + 2 \right)}} \)

Top: the graph of f = -log(log(x + 2)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -1/((x + 2)·log(x + 2)).

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. How we verify

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 log(x + 2) = 0 undefined where x + 2 = 0
6	✓ 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
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 + 2) = 0 undefined where 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: fail (style) — Step 1 lacks a rule label; it should be marked with "derivative" to indicate the application of the differentiation rule.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: fail (style) 2026-09-27 — Step 1 lacks a rule label; it should be marked with "derivative" to indicate the application of the differentiation rule.
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 2026-09-26
qwen3.6:27b-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 2026-09-27 with SymPy 1.14.0.