Derivative of \( \displaystyle 2 x \ln{\left(4 x + 1 \right)} - 2 x + \frac{\ln{\left(4 x + 1 \right)}}{2} \)
Problem 2.1437 · hard
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(4 x + 1 \right)} - 2 x + \frac{\ln{\left(4 x + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(2 x \ln{\left(4 x + 1 \right)} - 2 x + \frac{\ln{\left(4 x + 1 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(4 x + 1 \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 2 x \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(4 x + 1 \right)}}{2} - 2 \]constantDifferentiate the linear term 2*x.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \ln{\left(4 x + 1 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \frac{\ln{\left(4 x + 1 \right)}}{2} - 2 \]productApply the product rule to the first term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + 2 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(4 x + 1 \right)}}{2} - 2 \]derivativeDifferentiate 2*x.✓ Proved
- \[ = \frac{2 x \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} + 2 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(4 x + 1 \right)}}{2} - 2 \]chainApply the chain rule to log(4*x + 1).✓ Proved
- \[ = \frac{8 x}{4 x + 1} + 2 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(4 x + 1 \right)}}{2} - 2 \]derivative algebra algebraDifferentiate 4*x + 1. Simplify the fraction. Multiply 2*x by 4.✓ Proved
- \[ = \frac{8 x}{4 x + 1} + 2 \ln{\left(4 x + 1 \right)} + \frac{\frac{d}{d x} \ln{\left(4 x + 1 \right)}}{2} - 2 \]constant-multipleFactor out 1/2 from the last term.✓ Proved
- \[ = \frac{8 x}{4 x + 1} + 2 \ln{\left(4 x + 1 \right)} - 2 + \frac{\frac{d}{d x} \left(4 x + 1\right)}{2 \left(4 x + 1\right)} \]constant-multipleApply the constant multiple rule to the last term.✓ Proved
- \[ = \frac{8 x}{4 x + 1} + 2 \ln{\left(4 x + 1 \right)} - 2 + \frac{2}{4 x + 1} \]chain derivative algebraApply the chain rule to the last term. Differentiate 4*x + 1. Simplify the last term.✓ Proved
- \[ = 2 \ln{\left(4 x + 1 \right)} - 2 + \frac{8 x + 2}{4 x + 1} \]algebraCombine the fractions.✓ Proved
- \[ = 2 \ln{\left(4 x + 1 \right)} \]algebra algebra algebra simplifyFind a common denominator for the terms. Distribute the -2. Simplify the numerator. Final simplification.✓ Proved
Answer \( 2 \ln{\left(4 x + 1 \right)} \)
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.
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 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 19 | ✓ 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 12 is labeled "chain" but it merely applies an algebraic simplification; the label does not match the rule actually used.qwen3.6:27b-mlx: fail (error) — Step 11 is labeled 'constant-multiple' but actually applies the chain rule to the logarithm, which is a distinct differentiation rule. Step 12 is then labeled 'chain' but only performs algebraic substitution of the inner derivative, which was already handled in the previous step's logic or should have been part of the chain rule application in step 11. The separation of the chain rule into two steps with incorrect labels (calling the chain rule application 'constant-multiple') constitutes an error in rule naming.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 11 is labeled 'constant-multiple' but actually applies the chain rule to the logarithm, which is a distinct differentiation rule. Step 12 is then labeled 'chain' but only performs algebraic substitution of the inner derivative, which was already handled in the previous step's logic or should have been part of the chain rule application in step 11. The separation of the chain rule into two steps with incorrect labels (calling the chain rule application 'constant-multiple') constitutes an error in rule naming.gpt-oss:20b: fail (style) 2026-10-03 — Step 12 is labeled "chain" but it merely applies an algebraic simplification; the label does not match the rule actually used.qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.