Derivative of \( \displaystyle - \frac{\ln{\left(\ln{\left(5 x - 3 \right)} \right)}}{5} \)
Problem 2.1387 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(5 x - 3 \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\ln{\left(5 x - 3 \right)} \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\ln{\left(5 x - 3 \right)} \right)}}{5} \]constantPull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(5 x - 3 \right)}}{5 \ln{\left(5 x - 3 \right)}} \]chainApply the chain rule to the outer logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(5 x - 3\right)}{5 \left(5 x - 3\right) \ln{\left(5 x - 3 \right)}} \]chainApply the chain rule to the inner logarithm.✓ Proved
- \[ = - \frac{1}{\left(5 x - 3\right) \ln{\left(5 x - 3 \right)}} \]derivativeDifferentiate the innermost linear function.✓ Proved
- \[ = - \frac{1}{5 x \ln{\left(5 x - 3 \right)} - 3 \ln{\left(5 x - 3 \right)}} \]algebraSimplify the expression by multiplying the terms.✓ Proved
- \[ = - \frac{1}{\left(5 x - 3\right) \ln{\left(5 x - 3 \right)}} \]simplifyFactor the denominator.✓ Proved
Answer \( - \frac{1}{\left(5 x - 3\right) \ln{\left(5 x - 3 \right)}} \)
Mind the domain. The answer is also defined on (3/5, 4/5), 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 undefined where log(5*x - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(5*x - 3) = 0 undefined where 5*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(5*x - 3) = 0 undefined where 5*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(5*x - 3) = 0 undefined where 5*x - 3 = 0 undefined where 5*x*log(5*x - 3) - 3*log(5*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x*log(5*x - 3) - 3*log(5*x - 3) = 0 undefined where log(5*x - 3) = 0 undefined where 5*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(5*x - 3) = 0 undefined where 5*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 6 incorrectly multiplies the denominators: -1/5 * (1/log(...)) * (1/(5*x-3)) *5 simplifies to -1/(log(...)*(5*x-3)), not to the expression shown. The algebraic simplification is wrong.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — Step 6 incorrectly multiplies the denominators: -1/5 * (1/log(...)) * (1/(5*x-3)) *5 simplifies to -1/(log(...)*(5*x-3)), not to the expression shown. The algebraic simplification is wrong.qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — Step 6 incorrectly simplifies the product: -1/5*(1/log(5*x-3))*(1/(5*x-3))*5 should equal -1/(log(5*x-3)*(5*x-3)), but the solution writes -1/(5*x*log(5*x-3)-3*log(5*x-3)), which is algebraically incorrect.
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 2026-10-03 with SymPy 1.14.0.