Derivative of \( \displaystyle - 3 x + \left(3 x + \frac{3}{4}\right) \ln{\left(4 x + 1 \right)} \)
Problem 2.2294 · hard Video
Differentiate \( \displaystyle f(x) = - 3 x + \left(3 x + \frac{3}{4}\right) \ln{\left(4 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 3 x + \left(3 x + \frac{3}{4}\right) \ln{\left(4 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x + \frac{3}{4}\right) \ln{\left(4 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} 3 x \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(4 x + 1 \right)}}{4} \]productApply the product rule to the second term.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \ln{\left(4 x + 1 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \frac{3 \ln{\left(4 x + 1 \right)}}{4} \]productApply the product rule to the term 3*x*log(4*x + 1).✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + 3 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \frac{3 \ln{\left(4 x + 1 \right)}}{4} \]derivativeDifferentiate 3*x.✓ Proved
- \[ = \frac{3 x \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} + 3 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \frac{3 \ln{\left(4 x + 1 \right)}}{4} \]chainApply the chain rule to log(4*x + 1).✓ Proved
- \[ = \frac{12 x}{4 x + 1} + 3 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \frac{3 \ln{\left(4 x + 1 \right)}}{4} \]derivative algebraDifferentiate 4*x + 1. Simplify the product 3*x*4/ (4*x + 1).✓ Proved
- \[ = \frac{12 x}{4 x + 1} + 3 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3 \frac{d}{d x} \ln{\left(4 x + 1 \right)}}{4} \]algebraRemove unnecessary parentheses.✓ Proved
- \[ = \frac{12 x}{4 x + 1} + 3 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3 \frac{d}{d x} \left(4 x + 1\right)}{4 \left(4 x + 1\right)} \]chainApply the chain rule to the last term.✓ Proved
- \[ = \frac{12 x}{4 x + 1} + 3 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3}{4 x + 1} \]derivative algebraDifferentiate 4*x + 1. Simplify the constant multiplication.✓ Proved
- \[ = \frac{12 x}{4 x + 1} + 3 \ln{\left(4 x + 1 \right)} - 3 + \frac{3}{4 x + 1} \]derivativeDifferentiate -3*x.✓ Proved
- \[ = 3 \ln{\left(4 x + 1 \right)} - 3 + \frac{12 x + 3}{4 x + 1} \]algebra algebraCombine the fractions. Factor out 3 from the numerator.✓ Proved
- \[ = 3 \ln{\left(4 x + 1 \right)} \]simplify simplifyCancel the (4*x + 1) terms. Combine the constants.✓ Proved
Answer \( 3 \ln{\left(4 x + 1 \right)} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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 undefined where 4*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 17 | ✓ 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 incorrectly applies the product rule to the term (3*x + 3/4)*log(4*x + 1). The term is a product of a sum and a log, not a product of two single factors, so the step should first distribute the sum (an algebraic simplification) before applying the product rule. This violates the rule‑by‑step granularity requirement.qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly applies the product rule to split a sum term (3*x + 3/4)*log(...) into separate derivatives. The sum rule should have been applied to split the term inside the derivative, not the product rule. This violates the one-rule-per-step constraint and mislabels the operation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 3 incorrectly applies the product rule to split a sum term (3*x + 3/4)*log(...) into separate derivatives. The sum rule should have been applied to split the term inside the derivative, not the product rule. This violates the one-rule-per-step constraint and mislabels the operation.gpt-oss:20b: fail (error) 2026-10-17 — Step 3 incorrectly applies the product rule to the term (3*x + 3/4)*log(4*x + 1). The term is a product of a sum and a log, not a product of two single factors, so the step should first distribute the sum (an algebraic simplification) before applying the product rule. This violates the rule‑by‑step granularity requirement.qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 3 incorrectly applies the product rule to split a sum; the transition from step 2 to 3 is actually an application of the sum rule (linearity) to split (3*x + 3/4)*log(...) into two terms, not the product rule. The product rule is correctly applied in step 4.gpt-oss:20b: fail (style) 2026-10-15 — Step 3 incorrectly labels an algebraic distribution as a product rule. The step merely rewrites (3*x+3/4)*log(4*x+1) as 3*x*log(4*x+1)+3/4*log(4*x+1), which is an algebraic simplification, not a product rule application. The correct label should be "algebra".qwen3.6:27b-mlx: fail (error) 2026-10-14 — Step 3 incorrectly labels the expansion of a sum as the 'product' rule; the step applies the linearity of the derivative (sum rule) to split (3*x + 3/4)*log(...) into two separate derivative terms, not the product rule.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.