Home›Calculus 1›Product rule›Problem 2.2462

Derivative of \( \displaystyle - \frac{x^{2}}{2} + x + \left(x^{2} - 2 x + 1\right) \ln{\left(x - 1 \right)} \)

Problem 2.2462 · hard Video

Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x + \left(x^{2} - 2 x + 1\right) \ln{\left(x - 1 \right)} \).

\[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + x + \left(x^{2} - 2 x + 1\right) \ln{\left(x - 1 \right)}\right) \]
sumStart with the derivative of the entire function.✓ Proved
\[ = \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \left(x^{2} - 2 x + 1\right) \ln{\left(x - 1 \right)} \]
sumApply the sum rule to separate the terms.✓ Proved
\[ = \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \left(x - 1\right)^{2} \ln{\left(x - 1 \right)} \]
algebraSimplify the quadratic expression inside the parentheses.✓ Proved
\[ = \left(x - 1\right)^{2} \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)^{2} + \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]
productApply the product rule to the third term.✓ Proved
\[ = x + \ln{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)^{2} + \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) - 1 \]
logarithmicDifferentiate the natural logarithm term.✓ Proved
\[ = x + \left(2 x - 2\right) \ln{\left(x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) - 1 \]
chainApply the chain rule to the power term.✓ Proved
\[ = \left(2 x - 2\right) \ln{\left(x - 1 \right)} \]
power algebra algebraDifferentiate the power terms. Simplify the term (x-1)**2 / (x-1). Distribute the 2 in the product term.✓ Proved
\[ = 2 x \ln{\left(x - 1 \right)} - 2 \ln{\left(x - 1 \right)} \]
algebraCombine like terms and simplify.✓ Proved
\[ = \left(2 x - 2\right) \ln{\left(x - 1 \right)} \]
simplifyFinal simplified form.✓ Proved

Answer \( 2 \left(x - 1\right) \ln{\left(x - 1 \right)} \)

Top: the graph of f = -x²/2 + x + (x² - 2x + 1)·log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -x + (2x - 2)·log(x - 1) + 1 + (x² - 2x + 1)/(x - 1).

✓ 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. How we verify

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
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
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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies differentiation rules step-by-step. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed (sum, algebra, product, logarithmic, chain, power, simplify). The final result matches the stated answer.
gpt-oss:20b: fail (style) 2026-10-19 — Step 6 incorrectly labels the derivative of (x-1)^2 as a "chain" rule application; it should be the "power" rule. The labeling misrepresents the rule used, which could mislead a student.

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 2026-10-20 with SymPy 1.14.0.