Derivative of \( \displaystyle - x + \left(x - \frac{3}{5}\right) \ln{\left(5 x - 3 \right)} \)
Problem 2.2206 · hard Beautiful
Differentiate \( \displaystyle f(x) = - x + \left(x - \frac{3}{5}\right) \ln{\left(5 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(- x + \left(x - \frac{3}{5}\right) \ln{\left(5 x - 3 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x - \frac{3}{5}\right) \ln{\left(5 x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(x - \frac{3}{5}\right) \ln{\left(5 x - 3 \right)} - 1 \]constantDifferentiate the constant term.✓ Proved
- \[ = \left(x - \frac{3}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} \frac{d}{d x} \left(x - \frac{3}{5}\right) - 1 \]productApply the product rule.✓ Proved
- \[ = \left(x - \frac{3}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} - 1 \]derivative derivativeDifferentiate the first part of the product. Differentiate the constant part of the first term.✓ Proved
- \[ = \frac{\left(x - \frac{3}{5}\right) \frac{d}{d x} \left(5 x - 3\right)}{5 x - 3} + \ln{\left(5 x - 3 \right)} - 1 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \left(x - \frac{3}{5}\right)}{5 x - 3} + \ln{\left(5 x - 3 \right)} - 1 \]derivativeDifferentiate the inner function.✓ Proved
- \[ = \ln{\left(5 x - 3 \right)} \]algebra algebra algebra simplifySimplify the expression. Distribute the 5 in the numerator. Simplify the fraction. Final simplification.✓ Proved
Answer \( \ln{\left(5 x - 3 \right)} \)
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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0 |
| 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 (style) — Step 6 incorrectly labels a simplification as a derivative; the rule applied is "simplify", not "derivative".qwen3.6:27b-mlx: fail (style) — Step 3 is labeled 'constant' but applies the power rule to differentiate -x; the label should be 'derivative' or 'power'. Step 6 is labeled 'derivative' but performs algebraic simplification (removing the factor 1); the label should be 'algebra' or 'simplify'.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-13 — Step 6 incorrectly labels a simplification as a derivative; the rule applied is "simplify", not "derivative".qwen3.6:27b-mlx: fail (style) 2026-10-13 — Step 3 is labeled 'constant' but applies the power rule to differentiate -x; the label should be 'derivative' or 'power'. Step 6 is labeled 'derivative' but performs algebraic simplification (removing the factor 1); the label should be 'algebra' or 'simplify'.gpt-oss:20b: fail (error) 2026-10-13 — Step 3 incorrectly labels the derivative of -x as a constant; step 5 and 6 have misleading notes about differentiating constants when in fact they are differentiating non‑constant terms. These mislabelings violate the rule‑by‑step granularity requirement.qwen3.6:27b-mlx: fail (style) 2026-10-13 — Step 3 is labeled 'constant' but differentiates the term -x, which is not a constant; it should be labeled 'derivative' or 'power'. Step 6 is labeled 'derivative' but performs algebraic simplification (removing the factor 1), which should be labeled 'algebra' or 'simplify'.
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 2026-10-13 with SymPy 1.14.0.