Derivative of \( \displaystyle - 5 x + \left(5 x + \frac{5}{3}\right) \ln{\left(3 x + 1 \right)} \)
Problem 2.2201 · hard
Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x + \frac{5}{3}\right) \ln{\left(3 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 5 x + \left(5 x + \frac{5}{3}\right) \ln{\left(3 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x + \frac{5}{3}\right) \ln{\left(3 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(3 x + 1 \right)}}{3} \]productApply the product rule to the second term.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(3 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} \]constant-multipleApply the constant multiple rule to the third term.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} 5 x + \frac{d}{d x} \left(- 5 x\right) + \frac{5 \frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} \]productApply the product rule to the second term.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + 5 \ln{\left(3 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} - 5 \]derivative derivativeDifferentiate the first term. Differentiate the term 5*x.✓ Proved
- \[ = \frac{5 x \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} + 5 \ln{\left(3 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} - 5 \]chainApply the chain rule to the log term.✓ Proved
- \[ = \frac{15 x}{3 x + 1} + 5 \ln{\left(3 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} - 5 \]derivativeDifferentiate the inner function 3*x + 1.✓ Proved
- \[ = \frac{15 x}{3 x + 1} + 5 \ln{\left(3 x + 1 \right)} - 5 + \frac{5}{3 x + 1} \]chain algebraApply the chain rule to the third term. Simplify the terms.✓ Proved
- \[ = 5 \ln{\left(3 x + 1 \right)} - 5 + \frac{15 x + 5}{3 x + 1} \]algebra algebraCombine the fractions. Factor out 5 from the numerator.✓ Proved
- \[ = 5 \ln{\left(3 x + 1 \right)} \]simplify simplifyCancel the common factor (3*x + 1). Combine the constant terms.✓ Proved
Answer \( 5 \ln{\left(3 x + 1 \right)} \)
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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*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 + 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 + 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 + 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 + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*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 the product rule to the product (5*x+5/3)*log(3*x+1) but then immediately rewrites it as two separate products, effectively using both the product rule and the sum rule in one step. This violates the rule that each step must change only one thing. Additionally, step 7 incorrectly applies a derivative to the already differentiated term 5*x, applying two rules at once. These multi‑rule steps make the solution invalid.qwen3.6:27b-mlx: fail (error) — Step 3 claims to apply the product rule, but it actually applies the sum rule to split the term (5*x + 5/3)*log(3*x + 1) into two separate terms. The label 'product' is incorrect for a step that distributes a sum over a product or splits a sum; it should be labeled 'sum' or 'algebra'.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-13 — Step 3 applies the product rule to the product (5*x+5/3)*log(3*x+1) but then immediately rewrites it as two separate products, effectively using both the product rule and the sum rule in one step. This violates the rule that each step must change only one thing. Additionally, step 7 incorrectly applies a derivative to the already differentiated term 5*x, applying two rules at once. These multi‑rule steps make the solution invalid.qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 3 claims to apply the product rule, but it actually applies the sum rule to split the term (5*x + 5/3)*log(3*x + 1) into two separate terms. The label 'product' is incorrect for a step that distributes a sum over a product or splits a sum; it should be labeled 'sum' or 'algebra'.gpt-oss:20b: fail (error) 2026-10-13 — Step 3 applies two rules at once: it expands the product (5*x+5/3)*log(3*x+1) into 5*x*log(3*x+1)+(5/3)*log(3*x+1) and then applies a product rule to each term. The step is labeled "product", but the correct rule for the expansion is "sum" (or "algebra"), and the subsequent product rule is applied separately. Thus the step violates the one‑rule‑per‑step rule and is mis‑labelled.qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 3 applies the distributive property of the derivative (linearity/sum rule) to split a product term, but labels it 'product'. The product rule was not applied; the term was merely expanded into a sum of two terms. This violates the 'one rule per step' constraint and mislabels the operation.
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 2026-10-13 with SymPy 1.14.0.