Derivative of \( \displaystyle - 3 x + \left(3 x + \frac{3}{5}\right) \ln{\left(5 x + 1 \right)} \)
Problem 2.1750 · hard
Differentiate \( \displaystyle f(x) = - 3 x + \left(3 x + \frac{3}{5}\right) \ln{\left(5 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 3 x + \left(3 x + \frac{3}{5}\right) \ln{\left(5 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x + \frac{3}{5}\right) \ln{\left(5 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} 3 x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(5 x + 1 \right)}}{5} \]productApply the product rule to the second term.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \frac{3 \ln{\left(5 x + 1 \right)}}{5} \]productApply the product rule to the term 3*x*log(5*x + 1).✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 3 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \frac{3 \ln{\left(5 x + 1 \right)}}{5} \]derivativeDifferentiate 3*x.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 3 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3 \frac{d}{d x} \ln{\left(5 x + 1 \right)}}{5} \]constant-multipleApply the constant multiple rule to the third term.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 3 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3 \frac{d}{d x} \left(5 x + 1\right)}{5 \left(5 x + 1\right)} \]logarithmicDifferentiate the logarithm using the chain rule.✓ Proved
- \[ = \frac{15 x}{5 x + 1} + 3 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3 \frac{d}{d x} \left(5 x + 1\right)}{5 \left(5 x + 1\right)} \]chainApply the chain rule to the log argument.✓ Proved
- \[ = \frac{15 x}{5 x + 1} + 3 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3}{5 x + 1} \]derivative algebraDifferentiate 5*x + 1. Simplify the terms.✓ Proved
- \[ = \frac{15 x}{5 x + 1} + 3 \ln{\left(5 x + 1 \right)} - 3 + \frac{3}{5 x + 1} \]derivativeDifferentiate -3*x.✓ Proved
- \[ = 3 \ln{\left(5 x + 1 \right)} - 3 + \frac{15 x + 3}{5 x + 1} \]algebra algebraCombine the fractions. Factor the numerator.✓ Proved
- \[ = 3 \ln{\left(5 x + 1 \right)} \]simplify simplifySimplify the fraction. Combine the constants.✓ Proved
Answer \( 3 \ln{\left(5 x + 1 \right)} \)
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. A reviewer's objection to the explanation stands; see 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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 15 | ✓ 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 3 applies both the product rule and an algebraic rewrite in a single line, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'product' but performs an algebraic expansion (distributive property) to split a term into a sum, not the product rule for differentiation. Step 7 is labeled 'logarithmic' but the note claims it uses the chain rule; while the step does apply the chain rule, the label 'logarithmic' typically refers to the derivative of the log function itself, and the step actually combines the log derivative rule with the chain rule setup, but the primary defect is Step 3 mislabeling an algebraic split as a differentiation rule.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 3 applies both the product rule and an algebraic rewrite in a single line, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 3 is labeled 'product' but performs an algebraic expansion (distributive property) to split a term into a sum, not the product rule for differentiation. Step 7 is labeled 'logarithmic' but the note claims it uses the chain rule; while the step does apply the chain rule, the label 'logarithmic' typically refers to the derivative of the log function itself, and the step actually combines the log derivative rule with the chain rule setup, but the primary defect is Step 3 mislabeling an algebraic split as a differentiation rule.gpt-oss:20b: fail (error) 2026-10-07 — Step 3 incorrectly labels a distributive expansion as a product rule. The term (3*x + 3/5)*log(5*x + 1) is split into 3*x*log(5*x + 1) + (3/5)*log(5*x + 1) by algebra, not by the product rule. This violates the rule‑by‑step requirement.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 3 incorrectly applies the product rule to split a sum; the term (3*x + 3/5)*log(5*x + 1) is a product, but splitting it into two separate derivatives requires the sum rule, not the product rule. The product rule was correctly applied in Step 4 to the first part, but Step 3's label and logic are flawed because it treats the distribution of the derivative over the sum inside the product as a product rule application.
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 2026-10-07 with SymPy 1.14.0.