Derivative of \( \displaystyle - \frac{x^{2}}{2} + x \left(x - 2\right) \ln{\left(x - 1 \right)} + x + \ln{\left(x - 1 \right)} \)
Problem 2.154 · hard
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x \left(x - 2\right) \ln{\left(x - 1 \right)} + x + \ln{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + x \left(x - 2\right) \ln{\left(x - 1 \right)} + x + \ln{\left(x - 1 \right)}\right) \]sumStart with the derivative of the entire function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} \ln{\left(x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} \ln{\left(x - 1 \right)} + 1 \]constantThe derivative of x is 1.✓ Proved
- \[ = x \left(x - 2\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + x \ln{\left(x - 1 \right)} \frac{d}{d x} \left(x - 2\right) + \left(x - 2\right) \ln{\left(x - 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \ln{\left(x - 1 \right)} + 1 \]productApply the product rule to the second term.✓ Proved
- \[ = x \left(x - 2\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + x \ln{\left(x - 1 \right)} \frac{d}{d x} \left(x - 2\right) + \left(x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \ln{\left(x - 1 \right)} + 1 \]derivativeDifferentiate x.✓ Proved
- \[ = x \left(x - 2\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + x \ln{\left(x - 1 \right)} + \left(x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \ln{\left(x - 1 \right)} + 1 \]derivative algebraDifferentiate x - 2. Simplify the expression.✓ Proved
- \[ = x \left(x - 2\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + \left(2 x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \ln{\left(x - 1 \right)} + 1 \]algebraCombine log terms.✓ Proved
- \[ = \frac{x \left(x - 2\right)}{x - 1} + \left(2 x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \ln{\left(x - 1 \right)} + 1 \]derivativeDifferentiate log(x - 1).✓ Proved
- \[ = \left(2 x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \ln{\left(x - 1 \right)} + 1 + \frac{x^{2} - 2 x}{x - 1} \]algebraExpand the product.✓ Proved
- \[ = \left(2 x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + 1 + \frac{x^{2} - 2 x}{x - 1} + \frac{1}{x - 1} \]derivativeDifferentiate log(x - 1) again.✓ Proved
- \[ = - x + \left(2 x - 2\right) \ln{\left(x - 1 \right)} + 1 + \frac{x^{2} - 2 x}{x - 1} + \frac{1}{x - 1} \]derivativeDifferentiate -x**2/2.✓ Proved
- \[ = - x + \left(2 x - 2\right) \ln{\left(x - 1 \right)} + 1 + \frac{x^{2} - 2 x + 1}{x - 1} \]algebraCombine the fractions.✓ Proved
- \[ = \left(2 x - 2\right) \ln{\left(x - 1 \right)} \]algebra simplify algebra simplifyFactor the numerator. Simplify the fraction. Combine constants and x terms. Final simplification.✓ Proved
Answer \( 2 \left(x - 1\right) \log{\left(x - 1 \right)} \)
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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 17 | ✓ 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 4 applies the product rule to a product of three factors in one go, violating the one‑rule‑per‑step rule. Additionally, step 11 differentiates a log term that had already been differentiated in step 9, applying two rules at once and producing an extra 1/(x‑1).deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'constant', but it computes the derivative of x (which is 1), not a constant. The rule 'derivative' should have been used. Additionally, Step 11 applies the derivative rule to log(x-1) which was already differentiated in Step 9, resulting in a double differentiation error in the logic flow (though the algebraic equality holds due to previous cancellations, the step label and logic are flawed as it implies differentiating an already differentiated term again without context, or rather, it differentiates the term `Derivative(log(x-1), x)` which is not standard notation for a step; usually one evaluates the derivative. More critically, Step 3's label is incorrect for the operation performed).
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'constant', but it computes the derivative of x (which is 1), not a constant. The rule 'derivative' should have been used. Additionally, Step 11 applies the derivative rule to log(x-1) which was already differentiated in Step 9, resulting in a double differentiation error in the logic flow (though the algebraic equality holds due to previous cancellations, the step label and logic are flawed as it implies differentiating an already differentiated term again without context, or rather, it differentiates the term `Derivative(log(x-1), x)` which is not standard notation for a step; usually one evaluates the derivative. More critically, Step 3's label is incorrect for the operation performed).gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies the product rule to a product of three factors in one go, violating the one‑rule‑per‑step rule. Additionally, step 11 differentiates a log term that had already been differentiated in step 9, applying two rules at once and producing an extra 1/(x‑1).qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'constant', but it computes the derivative of x (which is 1), not a constant. The label 'derivative' should be used for applying d/dx to a basic function like x. Additionally, Step 11 repeats the differentiation of log(x-1) which was already handled in Step 9, resulting in a logical error where the term is differentiated twice or the state is inconsistent.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 4 applies the product rule to a product of three terms (x, (x-2), and log(x-1)) in a single step, violating the contract that each step must change only one thing (apply one rule to one term). The product rule is binary; differentiating a triple product requires two distinct steps or treating two factors as a single unit, which was not done.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 is labeled 'constant' but computes the derivative of x (which is 1, not 0); the label 'constant' implies the derivative of a constant is 0, which is incorrect here. Additionally, Step 4 applies the product rule to a three-factor term in a single step, violating the 'one rule per step' constraint.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 11 adds an extra 1/(x‑1) term, duplicating the derivative of log(x‑1). The correct expression after step 10 should be -x + (2*x‑2)*log(x‑1) + (x**2‑2*x)/(x‑1) + 1, without the additional 1/(x‑1).qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 3 is labeled 'constant' but computes the derivative of x (which is 1, not a constant derivative of 0). The label 'constant' applies to differentiating a constant term, not a linear term.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 9 (and step 10) incorrectly claim to differentiate log(x-1) but the expression still contains Derivative(log(x-1),x). The derivative should be replaced immediately, otherwise the subsequent algebraic simplifications are based on an incorrect intermediate expression.deepseek-r1:70b: pass 2026-09-17
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.