Derivative of \( \displaystyle - x + \left(x - \frac{1}{5}\right) \ln{\left(5 x - 1 \right)} \)
Problem 2.2083 · hard Beautiful
Differentiate \( \displaystyle f(x) = - x + \left(x - \frac{1}{5}\right) \ln{\left(5 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- x + \left(x - \frac{1}{5}\right) \ln{\left(5 x - 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x - \frac{1}{5}\right) \ln{\left(5 x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(x - \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \ln{\left(5 x - 1 \right)} \frac{d}{d x} \left(x - \frac{1}{5}\right) + \frac{d}{d x} \left(- x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = \left(x - \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \left(- x\right) \]derivative constant-multipleDifferentiate the term (x - 1/5). Simplify the constant multiplier.✓ Proved
- \[ = \frac{\left(x - \frac{1}{5}\right) \frac{d}{d x} \left(5 x - 1\right)}{5 x - 1} + \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \left(- x\right) \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \left(x - \frac{1}{5}\right)}{5 x - 1} + \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \left(- x\right) \]derivativeDifferentiate the inner function 5*x - 1.✓ Proved
- \[ = \ln{\left(5 x - 1 \right)} + \frac{d}{d x} \left(- x\right) + 1 \]algebra algebra algebraSimplify the product of the terms. Simplify the fraction (x - 1/5)*5/(5*x - 1) to (5x - 1)/(5x - 1). Simplify the fraction to 1.✓ Proved
- \[ = \ln{\left(5 x - 1 \right)} \]derivative simplifyDifferentiate the first term -x. Combine the constants -1 and 1.✓ Proved
Answer \( \ln{\left(5 x - 1 \right)} \)
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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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 |
| 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 8 applies multiple transformations at once and contains an incorrect intermediate expression. It rewrites (x-1/5)*5/(5x-1) as (5x-1)/(5x-1)*1/5*5/1, which is both algebraically incorrect and conflates several simplifications that should be split into separate steps.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications in steps 8-10 are valid and clearly labeled.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-11 — Step 8 applies multiple transformations at once and contains an incorrect intermediate expression. It rewrites (x-1/5)*5/(5x-1) as (5x-1)/(5x-1)*1/5*5/1, which is both algebraically incorrect and conflates several simplifications that should be split into separate steps.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications in steps 8-10 are valid and clearly labeled.gpt-oss:20b: fail (error) 2026-10-11 — Step 8 applies multiple algebraic simplifications at once and mis‑computes the product, leading to an incorrect intermediate expression. The subsequent steps incorrectly simplify the fraction, so the final result is not justified by the stated rules.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.
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 2026-10-11 with SymPy 1.14.0.