Derivative of \( \displaystyle \frac{\ln{\left(\ln{\left(2 x + 2 \right)} \right)}}{2} \)
Problem 2.239 · medium
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(2 x + 2 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\ln{\left(2 x + 2 \right)} \right)}}{2} \]constantStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\ln{\left(2 x + 2 \right)} \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(2 x + 2 \right)}}{2 \ln{\left(2 x + 2 \right)}} \]chainApply the chain rule to the outer logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(2 x + 2\right)}{2 \left(2 x + 2\right) \ln{\left(2 x + 2 \right)}} \]chainApply the chain rule to the inner logarithm.✓ Proved
- \[ = \frac{1}{\left(2 x + 2\right) \ln{\left(2 x + 2 \right)}} \]derivative algebra simplifyDifferentiate the innermost linear function. Simplify the product in the second term. Combine all terms into a single fraction.✓ Proved
Answer \( \frac{1}{\left(2 x + 2\right) \log{\left(2 x + 2 \right)}} \)
Mind the domain. The answer is also defined on (-1, -1/2), 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(2*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(2*x + 2) = 0 undefined where 2*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where log(2*x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where log(2*x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where log(2*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 2*x + 2 = 0 undefined where log(2*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 is labeled with the rule "constant", but no constant was extracted yet; the correct label should be "derivative" (the derivative of the whole expression).deepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, followed by two applications of the chain rule for the nested logarithms, and finally simplifies the algebraic expression. Each step isolates a single rule application as required.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, followed by two applications of the chain rule for the nested logarithms, and finally simplifies the algebraic expression. Each step isolates a single rule application as required.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 is labeled with the rule "constant", but no constant was extracted yet; the correct label should be "derivative" (the derivative of the whole expression).qwen3.6:27b-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: fail (style) 2026-09-19 — Step 1 incorrectly labels the operation as "constant"; it is simply taking the derivative of the whole expression, which should be labeled "derivative".qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 1 incorrectly labels the rule as 'constant' when it should be 'constant-multiple', potentially misleading a student about how constants are handled in differentiation.gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 1 incorrectly labels the rule as "constant" – the function is not a constant, it should be the definition of the derivative. Additionally, the domain restriction x > -0.5 (so that log(log(2x+2)) is defined) is omitted, which could mislead a student about where the derivative formula applies.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.