Derivative of \( \displaystyle 5 x \ln{\left(x - 3 \right)} - 5 x - 15 \ln{\left(x - 3 \right)} \)
Problem 2.261 · hard Beautiful
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(x - 3 \right)} - 5 x - 15 \ln{\left(x - 3 \right)} \).
- \[ \frac{d}{d x} \left(5 x \ln{\left(x - 3 \right)} - 5 x - 15 \ln{\left(x - 3 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = \frac{d}{d x} 5 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]constantThe derivative of 5*x is 5.✓ Proved
- \[ = 5 \frac{d}{d x} x \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]constant-multipleFactor out the constant 5 from the first term.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(x - 3 \right)} + 5 \ln{\left(x - 3 \right)} \frac{d}{d x} x - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]productApply the product rule to the first term.✓ Proved
- \[ = \frac{5 x}{x - 3} + 5 \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]chain algebraThe derivative of log(x - 3) is 1/(x - 3). Simplify the expression inside the parentheses.✓ Proved
- \[ = \frac{5 x}{x - 3} + 5 \ln{\left(x - 3 \right)} - 5 - \frac{15}{x - 3} \]chainThe derivative of 15*log(x - 3) is 15/(x - 3).✓ Proved
- \[ = 5 \ln{\left(x - 3 \right)} \]algebraCombine all terms over a common denominator.✓ Proved
- \[ = \frac{5 x \ln{\left(x - 3 \right)} - 15 \ln{\left(x - 3 \right)}}{x - 3} \]algebra simplifyExpand the numerator. Simplify the numerator by canceling terms.✓ Proved
- \[ = 5 \ln{\left(x - 3 \right)} \]algebra simplifyFactor out the common term 15*log(x - 3) is not correct, let's factor out 5*log(x-3). Cancel the common factor (x - 3).✓ Proved
Answer \( 5 \log{\left(x - 3 \right)} \)
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 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 3 incorrectly states that the derivative of 5*x is 5; the derivative of 5*x is 5, but the step label 'constant' is incorrect for differentiating a linear term (it should be 'derivative' or 'constant-multiple' combined with 'derivative', but primarily the note claims 'The derivative of 5*x is 5' which is true, but the rule 'constant' usually refers to d/dx(c)=0. More critically, Step 6 applies the chain rule to log(x-3) but fails to include the derivative of the inner function (which is 1) explicitly in the transformation from step 5 to 6, although the result is correct. However, the biggest error is in Step 3: the rule 'constant' is used to differentiate 5*x. The rule 'constant' typically implies d/dx(c) = 0. Differentiating 5*x requires the power rule or constant-multiple rule. Using 'constant' here is a mislabeling of the rule applied. Furthermore, Step 6's note says 'The derivative of log(x-3) is 1/(x-3)' which ignores the chain rule factor of 1 from the inner derivative, making the explanation misleading even if the algebra holds. But the most glaring defect is Step 3's label 'constant' for a non-constant term differentiation.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 incorrectly states that the derivative of 5*x is 5; the derivative of 5*x is 5, but the step label 'constant' is incorrect for differentiating a linear term (it should be 'derivative' or 'constant-multiple' combined with 'derivative', but primarily the note claims 'The derivative of 5*x is 5' which is true, but the rule 'constant' usually refers to d/dx(c)=0. More critically, Step 6 applies the chain rule to log(x-3) but fails to include the derivative of the inner function (which is 1) explicitly in the transformation from step 5 to 6, although the result is correct. However, the biggest error is in Step 3: the rule 'constant' is used to differentiate 5*x. The rule 'constant' typically implies d/dx(c) = 0. Differentiating 5*x requires the power rule or constant-multiple rule. Using 'constant' here is a mislabeling of the rule applied. Furthermore, Step 6's note says 'The derivative of log(x-3) is 1/(x-3)' which ignores the chain rule factor of 1 from the inner derivative, making the explanation misleading even if the algebra holds. But the most glaring defect is Step 3's label 'constant' for a non-constant term differentiation.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies the derivative rule to 5*x but labels it 'constant', which is incorrect; the derivative of 5*x is 5 via the power rule (or constant-multiple + derivative), not the constant rule. Additionally, Step 3 only differentiates one term while leaving others as 'Derivative(...)', violating the one-step-one-change principle if interpreted as a partial evaluation, but the primary defect is the mislabeling of the rule used to differentiate 5*x.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 incorrectly applies the 'constant' rule to the term 5*x; the derivative of 5*x is 5, which requires the constant-multiple rule (or power rule), not the rule for differentiating a constant. Additionally, Step 6 mislabels the differentiation of log(x-3) as 'chain' when it is primarily a 'logarithmic' derivative rule application, though the 'chain' label is acceptable for the inner function derivative, the primary error is in Step 3.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 is labeled 'constant' but computes the derivative of 5x (a linear term), which requires the power rule or derivative rule, not the constant rule. Step 6 is labeled 'chain' but performs two operations: differentiating log(u) and simplifying 1*log(x-3), violating the one-change-per-step constraint.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 12 note incorrectly states factoring out 15*log(x-3) instead of 5*log(x-3).gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17 — Step 12's note is slightly confusing but does not affect the correctness of the step.
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.