Home›Calculus 1›Derivatives of logarithms›Problem 2.2425

Derivative of \( \displaystyle 2 x \ln{\left(3 x - 3 \right)} - 2 x - 2 \ln{\left(x - 1 \right)} \)

Problem 2.2425 · hard Video

Differentiate \( \displaystyle f(x) = 2 x \ln{\left(3 x - 3 \right)} - 2 x - 2 \ln{\left(x - 1 \right)} \).

\[ \frac{d}{d x} \left(2 x \ln{\left(3 x - 3 \right)} - 2 x - 2 \ln{\left(x - 1 \right)}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 2 \ln{\left(x - 1 \right)} \]
sumApply the sum rule to differentiate each term separately.✓ Proved
\[ = \frac{d}{d x} 2 x \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 2 \ln{\left(x - 1 \right)} - 2 \]
constantThe derivative of 2*x is 2.✓ Proved
\[ = 2 x \frac{d}{d x} \ln{\left(3 x - 3 \right)} + \ln{\left(3 x - 3 \right)} \frac{d}{d x} 2 x - \frac{d}{d x} 2 \ln{\left(x - 1 \right)} - 2 \]
productApply the product rule to the first term.✓ Proved
\[ = 2 x \frac{d}{d x} \ln{\left(3 x - 3 \right)} + 2 \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 2 \ln{\left(x - 1 \right)} - 2 \]
derivativeThe derivative of 2*x is 2.✓ Proved
\[ = \frac{2 x \frac{d}{d x} \left(3 x - 3\right)}{3 x - 3} + 2 \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 2 \ln{\left(x - 1 \right)} - 2 \]
chainApply the chain rule to log(3*x - 3).✓ Proved
\[ = \frac{6 x}{3 x - 3} + 2 \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 2 \ln{\left(x - 1 \right)} - 2 \]
derivativeThe derivative of 3*x - 3 is 3.✓ Proved
\[ = \frac{6 x}{3 x - 3} + 2 \ln{\left(3 x - 3 \right)} - 2 - \frac{2 \frac{d}{d x} \left(x - 1\right)}{x - 1} \]
chainApply the chain rule to log(x - 1).✓ Proved
\[ = \frac{6 x}{3 x - 3} + 2 \ln{\left(3 x - 3 \right)} - 2 - \frac{2}{x - 1} \]
derivative algebra algebraThe derivative of x - 1 is 1. Simplify the products. Factor the denominator 3*x - 3.✓ Proved
\[ = \frac{2 x}{x - 1} + 2 \ln{\left(3 x - 3 \right)} - 2 - \frac{2}{x - 1} \]
algebraSimplify the fraction 6/3.✓ Proved
\[ = 2 \ln{\left(3 x - 3 \right)} - 2 + \frac{2 x - 2}{x - 1} \]
algebra algebraCombine the fractions with the same denominator. Factor the numerator.✓ Proved
\[ = 2 \ln{\left(3 x - 3 \right)} \]
algebra simplifySimplify the fraction (x-1)/(x-1). Final simplification.✓ Proved

Answer \( 2 \ln{\left(3 x - 3 \right)} \)

Top: the graph of f = 2x·log(3x - 3) - 2x - 2log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 6x/(3x - 3) + 2log(3x - 3) - 2 - 2/(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 undefined where 3*x - 3 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 3 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 3 = 0 undefined where x - 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 3 = 0 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 3*x - 3 = 0 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 3*x - 3 = 0 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 3*x - 3 = 0 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 undefined where x - 1 = 0
16	✓ 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 3 incorrectly labels the derivative of 2*x as "constant"; it should use the constant‑multiple rule. Step 5 also mislabels the derivative of 2*x as "derivative" instead of constant‑multiple. These labeling errors violate the single‑rule‑per‑step requirement.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19
gpt-oss:20b: fail (style) 2026-10-19 — Step 3 incorrectly labels the derivative of 2*x as "constant"; it should use the constant‑multiple rule. Step 5 also mislabels the derivative of 2*x as "derivative" instead of constant‑multiple. These labeling errors violate the single‑rule‑per‑step requirement.
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate and the logic is sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate and the logic is sound.
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.