Derivative of \( \displaystyle \frac{5 \ln{\left(\ln{\left(4 x \right)} \right)}}{4} \)

Problem 2.76 · medium

Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\ln{\left(4 x \right)} \right)}}{4} \).

\[ \frac{d}{d x} \frac{5 \ln{\left(\ln{\left(4 x \right)} \right)}}{4} \]
constantStart with the derivative of the function.✓ Proved
\[ = \frac{5 \frac{d}{d x} \ln{\left(\ln{\left(4 x \right)} \right)}}{4} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{5 \frac{d}{d x} \ln{\left(4 x \right)}}{4 \ln{\left(4 x \right)}} \]
chainApply the chain rule to the outer logarithm.✓ Proved
\[ = \frac{5 \frac{d}{d x} 4 x}{16 x \ln{\left(4 x \right)}} \]
chainApply the chain rule to the inner logarithm.✓ Proved
\[ = \frac{5}{4 x \ln{\left(4 x \right)}} \]
derivative algebra simplifyDifferentiate the innermost function. Cancel the constant 4. Simplify the final expression.✓ Proved

Answer \( \frac{5}{4 x \log{\left(4 x \right)}} \)

Mind the domain. The answer is also defined on (0, 1/4), 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(4*x) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(4*x) = 0 undefined where x = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 undefined where log(4*x) = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 undefined where log(4*x) = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 undefined where log(4*x) = 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 = 0 undefined where log(4*x) = 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 — The solution correctly applies the constant multiple rule and the chain rule in distinct steps. The algebraic simplification is handled in a separate step, adhering to the single-rule-per-step constraint.

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule and the chain rule in distinct steps. The algebraic simplification is handled in a separate step, adhering to the single-rule-per-step constraint.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, the chain rule for nested logarithms, and basic algebraic simplification. Each step isolates a single rule application as required.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule and the chain rule in distinct steps. The algebraic simplification is handled separately, adhering to the one-change-per-step constraint.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule and the chain rule in distinct steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: fail 2026-09-17 — The solution omits the domain restriction that x>1/4 (since log(log(4x)) requires 4x>1). This could mislead a student about where the derivative is valid.
deepseek-r1:70b: fail 2026-09-17 — The note in step 1 incorrectly attributes the action to the constant rule.

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.