Derivative of \( \displaystyle - 2 x + \left(2 x - 6\right) \ln{\left(x - 3 \right)} \)
Problem 2.1504 · hard Beautiful
Differentiate \( \displaystyle f(x) = - 2 x + \left(2 x - 6\right) \ln{\left(x - 3 \right)} \).
- \[ \frac{d}{d x} \left(- 2 x + \left(2 x - 6\right) \ln{\left(x - 3 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(2 x - 6\right) \ln{\left(x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(2 x - 6\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \left(2 x - 6\right) + \frac{d}{d x} \left(- 2 x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = \left(2 x - 6\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \left(2 x - 6\right) - 2 \]constantDifferentiate the first term.✓ Proved
- \[ = \left(2 x - 6\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + 2 \ln{\left(x - 3 \right)} - 2 \]derivativeDifferentiate the first part of the product rule.✓ Proved
- \[ = 2 \ln{\left(x - 3 \right)} - 2 + \frac{2 x - 6}{x - 3} \]derivative algebraDifferentiate the logarithm. Factor the term (2*x - 6).✓ Proved
- \[ = 2 \ln{\left(x - 3 \right)} \]simplify simplifySimplify the fraction. Combine the constant terms.✓ Proved
Answer \( 2 \ln{\left(x - 3 \right)} \)
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 |
| 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 7 combines factoring of (2*x-6) into 2*(x-3) and cancellation with the denominator in one line, applying two algebraic rules at once, which violates the one‑change‑per‑step rule.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.gpt-oss:20b: fail (error) 2026-10-04 — Step 7 combines factoring of (2*x-6) into 2*(x-3) and cancellation with the denominator in one line, applying two algebraic rules at once, which violates the one‑change‑per‑step rule.qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.