Derivative of \( \displaystyle x \ln{\left(x - 3 \right)} - x - 3 \ln{\left(x - 3 \right)} \)

Problem 2.90 · hard Beautiful

Differentiate \( \displaystyle f(x) = x \ln{\left(x - 3 \right)} - x - 3 \ln{\left(x - 3 \right)} \).

\[ \frac{d}{d x} \left(x \ln{\left(x - 3 \right)} - x - 3 \ln{\left(x - 3 \right)}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 3 \right)} \]
sumApply the sum rule for differentiation.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 3 \right)} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} x - \frac{d}{d x} 3 \ln{\left(x - 3 \right)} - 1 \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} - 3 \frac{d}{d x} \ln{\left(x - 3 \right)} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = \frac{x}{x - 3} + \ln{\left(x - 3 \right)} - 1 - \frac{3}{x - 3} \]
logarithmic constant-multipleThe derivative of log(x - 3) is 1/(x - 3). Simplify the coefficients.✓ Proved
\[ = \ln{\left(x - 3 \right)} \]
algebra simplify simplifyCombine the fractions with the same denominator. Simplify the fraction (x-3)/(x-3). Final simplification.✓ Proved

Answer \( \log{\left(x - 3 \right)} \)

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 undefined where x - 3 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 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
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
deepseek-r1:70b: pass
qwen3.6:27b-mlx: fail (error) — Step 6 incorrectly applies the 'logarithmic' rule to the term 3*log(x-3) without first factoring out the constant 3. The step simultaneously removes the derivative operator from the coefficient 3 (applying constant-multiple) and differentiates the log function (applying logarithmic/chain), violating the one-rule-per-step constraint. Additionally, the note claims the derivative of log(x-3) is 1/(x-3), ignoring the chain rule factor of 1, which is mathematically incorrect for the inner function (x-3).

Every verdict on record (12)

qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 incorrectly applies the 'logarithmic' rule to the term 3*log(x-3) without first factoring out the constant 3. The step simultaneously removes the derivative operator from the coefficient 3 (applying constant-multiple) and differentiates the log function (applying logarithmic/chain), violating the one-rule-per-step constraint. Additionally, the note claims the derivative of log(x-3) is 1/(x-3), ignoring the chain rule factor of 1, which is mathematically incorrect for the inner function (x-3).
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 incorrectly applies the logarithmic rule by omitting the chain rule factor (the derivative of the inner function x-3 is 1, but the step label 'logarithmic' implies a direct application without accounting for the inner derivative, or rather, the note claims the derivative of log(x-3) is 1/(x-3) which is technically correct only because the inner derivative is 1, but the step combines the evaluation of the derivative of the outer function and the inner function into one step labeled 'logarithmic' while the previous steps separated product and constant rules. More critically, Step 5 applies 'constant' to evaluate Derivative(x,x) but leaves Derivative(3*log(x-3), x) as 3*Derivative(log(x-3), x) in the same step? No, Step 5 evaluates Derivative(x,x) to 1 AND pulls out the constant 3 from the last term. Applying two rules (derivative of x and constant multiple) in one step is a defect.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 applies two rules simultaneously: it differentiates x to 1 (derivative rule) and factors out the constant 3 from the last term (constant-multiple rule). The contract requires each step to change only one thing.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 5 applies two rules at once (replacing Derivative(x,x) with 1 and converting Derivative(3*log(x-3),x) to 3*Derivative(log(x-3),x)). It also labels the step only as "constant", which does not name the constant‑multiple rule applied to the second term.
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 applies two rules simultaneously: it evaluates Derivative(x, x) to 1 and factors out the constant 3 from Derivative(3*log(x - 3), x). The contract requires each step to change only one thing.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: fail 2026-09-17 — Step 5 applies more than one rule (derivative of x and constant‑multiple rule) but the note only mentions the derivative of x, misleading the student. The rule label "constant" is also incomplete for the constant‑multiple operation performed.
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.