Home›Calculus 1›Chain rule›Problem 2.105

Derivative of \( \displaystyle 5 \ln{\left(\ln{\left(x + 2 \right)} \right)} \)

Problem 2.105 · medium

Differentiate \( \displaystyle f(x) = 5 \ln{\left(\ln{\left(x + 2 \right)} \right)} \).

\[ \frac{d}{d x} 5 \ln{\left(\ln{\left(x + 2 \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 5 \frac{d}{d x} \ln{\left(\ln{\left(x + 2 \right)} \right)} \]
constant-multiple chainPull the constant out of the derivative. Apply the chain rule to the outer logarithm.✓ Proved
\[ = \frac{5 \frac{d}{d x} \ln{\left(x + 2 \right)}}{\ln{\left(x + 2 \right)}} \]
chainApply the chain rule to the inner logarithm.✓ Proved
\[ = \frac{5}{\left(x + 2\right) \ln{\left(x + 2 \right)}} \]
chain simplifyDifferentiate the innermost term. Simplify the expression.✓ Proved

Answer \( \frac{5}{\left(x + 2\right) \log{\left(x + 2 \right)}} \)

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.

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. The reviewers disagree about how one step is explained; every verdict is in the receipt.

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(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 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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (15)

qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 3 applies no change yet labels it as a chain rule application, and step 4 combines two chain rule applications in one line. Each step must alter only one rule, so the solution violates the contract.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (style) 2026-09-19 — Step 3 adds parentheses without applying any rule; it should be an algebraic or rewrite step, not a chain rule application.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: fail 2026-09-17 — Step 3 incorrectly claims to apply the chain rule but does not change the expression, and step 5 mislabels the derivative of log as a chain rule; domain restrictions (x > -1) are omitted.
deepseek-r1:70b: fail 2026-09-17 — The solution ignores the domain restriction x > -1, which is necessary for the logarithm arguments to be positive.

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.