Derivative of \( \displaystyle \frac{3 \ln{\left(\ln{\left(2 x - 3 \right)} \right)}}{2} \)
Problem 2.642 · hard
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\ln{\left(2 x - 3 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{3 \ln{\left(\ln{\left(2 x - 3 \right)} \right)}}{2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\ln{\left(2 x - 3 \right)} \right)}}{2} \]constant-multiple chainPull out the constant factor. Apply the chain rule to the outer logarithm.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2 \ln{\left(2 x - 3 \right)}} \]chainApply the chain rule to the inner logarithm.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(2 x - 3\right)}{2 \left(2 x - 3\right) \ln{\left(2 x - 3 \right)}} \]chainApply the chain rule to the innermost linear term.✓ Proved
- \[ = \frac{3}{\left(2 x - 3\right) \ln{\left(2 x - 3 \right)}} \]derivative algebra simplifyDifferentiate the linear term. Multiply the terms in the numerator and denominator. Simplify the constant factors.✓ Proved
Answer \( \frac{3}{\left(2 x - 3\right) \log{\left(2 x - 3 \right)}} \)
Mind the domain. The answer is also defined on (3/2, 2), 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(2*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(2*x - 3) = 0 undefined where 2*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(2*x - 3) = 0 undefined where 2*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(2*x - 3) = 0 undefined where 2*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(2*x - 3) = 0 undefined where 2*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(2*x - 3) = 0 undefined where 2*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: fail (error) — Step 5 incorrectly labels the differentiation of the linear term "2*x - 3" as a chain rule application. The derivative of a linear function is a constant‑multiple rule, not a chain rule. The step should be labeled "derivative" (or "constant-multiple" if pulling out the 2).qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 5 incorrectly labels the differentiation of the linear term "2*x - 3" as a chain rule application. The derivative of a linear function is a constant‑multiple rule, not a chain rule. The step should be labeled "derivative" (or "constant-multiple" if pulling out the 2).qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 5 incorrectly labels the application of the derivative of the linear term as a "chain" rule. The derivative of 2*x-3 is 2, which requires only the "derivative" rule, not "chain". This mislabeling could mislead a student about when the chain rule applies.
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.