Derivative of \( \displaystyle - 5 x + \left(5 x + \frac{5}{2}\right) \ln{\left(2 x + 1 \right)} \)
Problem 2.2187 · hard
Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x + \frac{5}{2}\right) \ln{\left(2 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 5 x + \left(5 x + \frac{5}{2}\right) \ln{\left(2 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x + \frac{5}{2}\right) \ln{\left(2 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} \]productApply the product rule to the second term.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(2 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} \]constant-multipleFactor out the constant 5/2.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} 5 x + \frac{d}{d x} \left(- 5 x\right) + \frac{5 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} \]productApply the product rule to 5*x*log(2*x + 1).✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 5 \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{5 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} \]derivativeDifferentiate 5*x.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 5 \ln{\left(2 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 5 \]derivativeDifferentiate -5*x.✓ Proved
- \[ = \frac{5 x \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} + 5 \ln{\left(2 x + 1 \right)} - 5 + \frac{5 \frac{d}{d x} \left(2 x + 1\right)}{2 \left(2 x + 1\right)} \]chainApply the chain rule to log(2*x + 1).✓ Proved
- \[ = \frac{10 x}{2 x + 1} + 5 \ln{\left(2 x + 1 \right)} - 5 + \frac{5}{2 x + 1} \]derivative algebraDifferentiate 2*x + 1. Simplify the terms.✓ Proved
- \[ = 5 \ln{\left(2 x + 1 \right)} - 5 + \frac{10 x + 5}{2 x + 1} \]algebra algebraCombine the fractions. Factor out 5 from the numerator.✓ Proved
- \[ = 5 \ln{\left(2 x + 1 \right)} \]algebra simplifySimplify the fraction. Final simplification.✓ Proved
Answer \( 5 \ln{\left(2 x + 1 \right)} \)
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 |
| 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 2*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 14 | ✓ 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 3 incorrectly labels the distributive expansion of (5*x+5/2)*log(2*x+1) as a product rule. The correct label would be "algebra" or "rewrite", not "product".qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly labels the expansion of a sum as the 'product' rule; it should be 'sum'. Step 8 incorrectly labels the application of the logarithmic derivative formula as 'chain'; it should be 'logarithmic' (or 'derivative' if 'logarithmic' is not accepted, but 'chain' is incorrect because the chain rule is applied in the subsequent step 9).
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-13 — Step 3 incorrectly labels the distributive expansion of (5*x+5/2)*log(2*x+1) as a product rule. The correct label would be "algebra" or "rewrite", not "product".qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 3 incorrectly labels the expansion of a sum as the 'product' rule; it should be 'sum'. Step 8 incorrectly labels the application of the logarithmic derivative formula as 'chain'; it should be 'logarithmic' (or 'derivative' if 'logarithmic' is not accepted, but 'chain' is incorrect because the chain rule is applied in the subsequent step 9).gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 3 claims to apply the product rule, but it actually applies the sum rule to split (5*x + 5/2)*log(2*x + 1) into 5*x*log(2*x + 1) + 5/2*log(2*x + 1). The product rule differentiates a product; it does not distribute a sum over a factor.
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.