Derivative of \( \displaystyle - 5 x + \left(5 x - \frac{5}{2}\right) \ln{\left(2 x - 1 \right)} \)
Problem 2.2202 · 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
- \[ = \left(5 x - \frac{5}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(5 x - \frac{5}{2}\right) + \frac{d}{d x} \left(- 5 x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = - \left(5 x - \frac{5}{2}\right) \ln{\left(2 x - 1 \right)} \frac{d}{d x} \frac{1}{2} + \left(5 x - \frac{5}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} 5 x + \frac{d}{d x} \left(- 5 x\right) \]sumDifferentiate the terms inside the product rule.✓ Proved
- \[ = - \left(5 x - \frac{5}{2}\right) \ln{\left(2 x - 1 \right)} \frac{d}{d x} \frac{5}{2} + \left(5 x - \frac{5}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} 5 x + \frac{d}{d x} \left(- 5 x\right) \]constantDifferentiate the constant term 5/2.✓ Proved
- \[ = - \left(5 x - \frac{5}{2}\right) \ln{\left(2 x - 1 \right)} \frac{d}{d x} \frac{5}{2} + \left(5 x - \frac{5}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + 5 \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) \]constant-multipleDifferentiate 5*x.✓ Proved
- \[ = \left(5 x - \frac{5}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + 5 \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) \]constant-multiple algebraDifferentiate the constant 5/2. Simplify the zero term.✓ Proved
- \[ = 5 \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{\left(5 x - \frac{5}{2}\right) \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = 5 \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{2 \left(5 x - \frac{5}{2}\right)}{2 x - 1} \]constant-multiple algebraDifferentiate 2*x - 1. Simplify the expression.✓ Proved
- \[ = 5 \ln{\left(2 x - 1 \right)} - 5 + \frac{2 \left(5 x - \frac{5}{2}\right)}{2 x - 1} \]derivativeEvaluate the derivative of -5*x.✓ Proved
- \[ = 5 \ln{\left(2 x - 1 \right)} - 5 + \frac{10 x - 5}{2 x - 1} \]algebra algebraDistribute the 2 in the numerator. Factor out 5 from the numerator.✓ Proved
- \[ = 5 \ln{\left(2 x - 1 \right)} \]simplify algebraCancel the common factor (2*x - 1). Combine the constant terms.✓ 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. 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 |
| 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 undefined where 2*x - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 16 | ✓ 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) — Several steps apply multiple rules at once and mislabel the operations. For example, step 4 expands the product rule but also differentiates inside the sum, and step 5 incorrectly applies the constant rule to a non‑constant term. These combined operations violate the one‑rule‑per‑step requirement and produce an incorrect derivative sequence.qwen3.6:27b-mlx: fail (error) — Step 4 is mathematically incorrect; it attempts to split the derivative of the product term using a 'sum' rule but produces an invalid expression involving `Derivative(1/2, x)` and incorrectly distributes terms. Step 5 also contains a logical error by differentiating a constant `5/2` as if it were a variable term in a product, leading to nonsensical intermediate steps.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-13 — Several steps apply multiple rules at once and mislabel the operations. For example, step 4 expands the product rule but also differentiates inside the sum, and step 5 incorrectly applies the constant rule to a non‑constant term. These combined operations violate the one‑rule‑per‑step requirement and produce an incorrect derivative sequence.qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 4 is mathematically incorrect; it attempts to split the derivative of the product term using a 'sum' rule but produces an invalid expression involving `Derivative(1/2, x)` and incorrectly distributes terms. Step 5 also contains a logical error by differentiating a constant `5/2` as if it were a variable term in a product, leading to nonsensical intermediate steps.gpt-oss:20b: fail (error) 2026-10-13 — Step 4 incorrectly applies the product rule: it splits the derivative of (5*x-5/2) into a derivative of 5*x and a derivative of 1/2, introducing an extra term -(5*x-5/2)*Derivative(1/2,x)*log(2*x-1). The derivative of (5*x-5/2) is simply 5, so the extra term is wrong and propagates errors through subsequent steps.qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 4 is mathematically incorrect; it attempts to distribute the derivative operator over the sum inside the product rule term but incorrectly introduces a negative sign and a derivative of a constant term that does not exist in the original expression. Specifically, Derivative(5*x - 5/2, x) is simply 5, not a sum involving Derivative(1/2, x) with a negative sign attached to the log term in that manner. The step violates the 'one rule per step' constraint by mixing differentiation rules with incorrect algebraic manipulation.
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.