Home›Calculus 1›Derivatives of logarithms›Problem 2.2394

Derivative of \( \displaystyle - 3 x + \left(3 x - 9\right) \ln{\left(x - 3 \right)} \)

Problem 2.2394 · hard Beautiful Video

Differentiate \( \displaystyle f(x) = - 3 x + \left(3 x - 9\right) \ln{\left(x - 3 \right)} \).

\[ \frac{d}{d x} \left(- 3 x + \left(3 x - 9\right) \ln{\left(x - 3 \right)}\right) \]
sumStart with the derivative of the entire function.✓ Proved
\[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x - 9\right) \ln{\left(x - 3 \right)} \]
sumApply the sum rule to separate the terms.✓ Proved
\[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x \ln{\left(x - 3 \right)} - 9 \ln{\left(x - 3 \right)}\right) \]
algebraDistribute the term (3*x - 9) into the logarithm.✓ Proved
\[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} 3 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 9 \ln{\left(x - 3 \right)} \]
sumApply the sum rule to the expanded expression.✓ Proved
\[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} 3 x \ln{\left(x - 3 \right)} - 9 \frac{d}{d x} \ln{\left(x - 3 \right)} \]
constantPull out the constant factor 9.✓ Proved
\[ = \frac{d}{d x} 3 x \ln{\left(x - 3 \right)} - 9 \frac{d}{d x} \ln{\left(x - 3 \right)} - 3 \]
constantThe derivative of -3*x is -3.✓ Proved
\[ = 3 x \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} 3 x - 9 \frac{d}{d x} \ln{\left(x - 3 \right)} - 3 \]
productApply the product rule to the term 3*x*log(x - 3).✓ Proved
\[ = 3 x \frac{d}{d x} \ln{\left(x - 3 \right)} + 3 \ln{\left(x - 3 \right)} - 9 \frac{d}{d x} \ln{\left(x - 3 \right)} - 3 \]
derivativeThe derivative of 3*x is 3.✓ Proved
\[ = \frac{3 x}{x - 3} + 3 \ln{\left(x - 3 \right)} - 3 - \frac{9}{x - 3} \]
logarithmicThe derivative of log(x - 3) is 1/(x - 3).✓ Proved
\[ = 3 \ln{\left(x - 3 \right)} - 3 + \frac{3 x - 9}{x - 3} \]
algebra algebraCombine the terms with the common denominator (x - 3). Factor out 3 from the numerator.✓ Proved
\[ = 3 \ln{\left(x - 3 \right)} \]
algebra simplifySimplify the fraction (x - 3)/(x - 3) to 1. Combine the constant terms -3 and 3.✓ Proved

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

Top: the graph of f = -3x + (3x - 9)·log(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3log(x - 3) - 3 + (3x - 9)/(x - 3).

✓ 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 undefined where x - 3 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0
13	✓ 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 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels used are appropriate for the operations performed.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.