Home›Calculus 1›Quotient rule›Problem 2.642

Derivative of \( \displaystyle \frac{3 \ln{\left(\ln{\left(2 x - 3 \right)} \right)}}{2} \)

Problem 2.642 · hard Video

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 \( \frac{3}{\left(2 x - 3\right) \log{\left(2 x - 3 \right)}} \)

Top: the graph of f = 3log(log(2x - 3))/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3/((2x - 3)·log(2x - 3)).

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.

✓ 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
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(2x - 3) = 0 undefined where 2x - 3 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2x - 3 = 0 undefined where log(2x - 3) = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2x - 3 = 0 undefined where log(2x - 3) = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2x - 3 = 0 undefined where log(2x - 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 2x - 3 = 0 undefined where log(2x - 3) = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-11: fail — The math is correct and every line is SymPy-proved, including the final answer. However, line 3 is line 2 with only extra parentheses, yet it is labeled 'chain' and described as applying the chain rule. The outer-log chain rule is actually applied in line 4, so line 3 is a redundant no-op step with a misleading label.

Every verdict on record (7)

claude-sonnet-5-5: fail (style) 2026-10-11 — The math is correct and every line is SymPy-proved, including the final answer. However, line 3 is line 2 with only extra parentheses, yet it is labeled 'chain' and described as applying the chain rule. The outer-log chain rule is actually applied in line 4, so line 3 is a redundant no-op step with a misleading label.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-21
gpt-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 2026-09-21
gpt-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.