Derivative of \( \displaystyle 3 x \left(\ln{\left(5 x \right)} - 1\right) \)
Problem 2.491 · medium Video
Differentiate \( \displaystyle f(x) = 3 x \left(\ln{\left(5 x \right)} - 1\right) \).
- \[ \frac{d}{d x} 3 x \left(\ln{\left(5 x \right)} - 1\right) \]Start with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} x \left(\ln{\left(5 x \right)} - 1\right) \]constant-multiplePull out the constant factor 3.✓ Proved
- \[ = 3 x \frac{d}{d x} \left(\ln{\left(5 x \right)} - 1\right) + 3 \left(\ln{\left(5 x \right)} - 1\right) \frac{d}{d x} x \]productApply the product rule to the remaining term.✓ Proved
- \[ = 3 x \left(- \frac{d}{d x} 1 + \frac{d}{d x} \ln{\left(5 x \right)}\right) + 3 \ln{\left(5 x \right)} - 3 \]algebraDistribute the derivative across the subtraction.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(5 x \right)} + 3 \ln{\left(5 x \right)} - 3 \]constant algebraThe derivative of the constant 1 is 0. Simplify the expression inside the parentheses.✓ Proved
- \[ = 3 \ln{\left(5 x \right)} \]logarithmic algebra algebra simplifyApply the chain rule to the logarithm. Simplify the term inside the parentheses. Simplify the term x/x. Combine the constant terms.✓ Proved
Answer \( 3 \log{\left(5 x \right)} \)
✓ 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 |
| 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: fail (style) — Step 1 lacks a label (should be "derivative"). Step 4 applies the linearity of the derivative, not an algebraic simplification, so the label "algebra" is incorrect.qwen3.6:27b-mlx: fail (misleading) — Step 7 is labeled 'logarithmic' but the note explicitly states 'Apply the chain rule'. The step actually applies both the logarithmic derivative rule and the chain rule, violating the single-rule constraint and mislabeling the primary operation described in the note.
Senior review claude-sonnet-5-5, 2026-10-09: pass — Every line was proved by SymPy, and the derivation is correct: 3*(log(5x) - 1 + 1) = 3*log(5x). The labels are acceptable under the contract, and the remaining objections are only about wording.
gpt-oss:20b: dismiss — Line 1 is just the problem statement and needs no label. In step 4, distributing the derivative over a subtraction is the sum rule, but the contract says that when two labels fit either is acceptable, and 'algebra' is a defensible name for that distribution, so it is not a wrong label.qwen3.6:27b-mlx: dismiss — Differentiating log(5x) as 1/(5x)*5 uses the logarithmic derivative together with the chain rule's inner derivative of 5x. The contract counts evaluating a simple inner derivative as part of one chain-rule step, so this is a single step and 'logarithmic' is a legitimate label.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-09 — Every line was proved by SymPy, and the derivation is correct: 3*(log(5x) - 1 + 1) = 3*log(5x). The labels are acceptable under the contract, and the remaining objections are only about wording.qwen3.6:27b-mlx: fail (misleading) 2026-09-21 — Step 7 is labeled 'logarithmic' but the note explicitly states 'Apply the chain rule'. The step actually applies both the logarithmic derivative rule and the chain rule, violating the single-rule constraint and mislabeling the primary operation described in the note.gpt-oss:20b: fail (style) 2026-09-21 — Step 1 lacks a label (should be "derivative"). Step 4 applies the linearity of the derivative, not an algebraic simplification, so the label "algebra" is incorrect.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 7 is labeled 'logarithmic' but the note explicitly states 'Apply the chain rule', and the step performs the chain rule differentiation of log(5x). The label should be 'chain' (or 'logarithmic' with a corrected note), but the mismatch between label and note is a defect in clarity. Additionally, Step 4 is labeled 'algebra' but performs the linearity of the derivative operator (distributing derivative over subtraction), which is typically covered by 'sum' or 'derivative' rules in this vocabulary, though 'algebra' is sometimes used loosely for linearity. However, the primary issue is the label/note mismatch in Step 7.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 lacks a proper rule label. It should be labeled "derivative" instead of "unlabelled".qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 7 is labeled 'logarithmic' but the note explicitly states 'Apply the chain rule', and the step performs both the logarithmic derivative and the chain rule multiplication. This violates the single-rule-per-step constraint and mislabels the operation.gpt-oss:20b: pass 2026-09-20
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.