Derivative of \( \displaystyle - 5 x + \left(5 x - \frac{5}{4}\right) \ln{\left(4 x - 1 \right)} \)
Problem 2.2205 · hard
Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x - \frac{5}{4}\right) \ln{\left(4 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- 5 x + \left(5 x - \frac{5}{4}\right) \ln{\left(4 x - 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x - \frac{5}{4}\right) \ln{\left(4 x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(5 x - \frac{5}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} \left(5 x - \frac{5}{4}\right) + \frac{d}{d x} \left(- 5 x\right) \]productApply the product rule.✓ Proved
- \[ = \left(5 x - \frac{5}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \left(- \frac{d}{d x} \frac{5}{4} + \frac{d}{d x} 5 x\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) \]algebraDistribute the derivative over the subtraction.✓ Proved
- \[ = \left(5 - \frac{d}{d x} \frac{5}{4}\right) \ln{\left(4 x - 1 \right)} + \left(5 x - \frac{5}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) \]constant-multipleDifferentiate the linear term.✓ Proved
- \[ = \left(5 x - \frac{5}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + 5 \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) \]constant algebraThe derivative of a constant is zero. Simplify the expression.✓ Proved
- \[ = 5 \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{\left(5 x - \frac{5}{4}\right) \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = 5 \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{\left(5 x - \frac{5}{4}\right) \left(- \frac{d}{d x} 1 + \frac{d}{d x} 4 x\right)}{4 x - 1} \]sumDifferentiate the inner function.✓ Proved
- \[ = 5 \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{4 \left(5 x - \frac{5}{4}\right)}{4 x - 1} \]constant algebraThe derivative of the constant 1 is zero. Simplify the product.✓ Proved
- \[ = 5 \ln{\left(4 x - 1 \right)} - 5 + \frac{4 \left(5 x - \frac{5}{4}\right)}{4 x - 1} \]derivativeEvaluate the remaining derivative.✓ Proved
- \[ = 5 \ln{\left(4 x - 1 \right)} - 5 + \frac{20 x - 5}{4 x - 1} \]algebra algebraMultiply the numerator. Factor out 5 from the numerator.✓ Proved
- \[ = 5 \ln{\left(4 x - 1 \right)} \]simplify algebraCancel the common term in the fraction. Combine the constant terms.✓ Proved
Answer \( 5 \ln{\left(4 x - 1 \right)} \)
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 |
| 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 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 |
| 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 applies the product rule and simultaneously rewrites the first factor (5*x - 5/4) as (5*x - 5*5/4/5). This is two operations in one step, violating the rule that each step must change only one thing. The step should be split into a product rule application followed by a separate simplification step.qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly applies the product rule by replacing the derivative of the first factor with a derivative of an algebraically modified expression (5*x - 5*5/4/5) rather than the original term (5*x - 5/4). While the algebraic value is identical, the step violates the 'one thing' rule by performing an algebraic rewrite inside the derivative operator simultaneously with applying the product rule structure, and the label 'product' does not account for the internal algebraic manipulation of the term being differentiated.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-13 — Step 3 applies the product rule and simultaneously rewrites the first factor (5*x - 5/4) as (5*x - 5*5/4/5). This is two operations in one step, violating the rule that each step must change only one thing. The step should be split into a product rule application followed by a separate simplification step.qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 3 incorrectly applies the product rule by replacing the derivative of the first factor with a derivative of an algebraically modified expression (5*x - 5*5/4/5) rather than the original term (5*x - 5/4). While the algebraic value is identical, the step violates the 'one thing' rule by performing an algebraic rewrite inside the derivative operator simultaneously with applying the product rule structure, and the label 'product' does not account for the internal algebraic manipulation of the term being differentiated.gpt-oss:20b: fail (error) 2026-10-13 — Step 3 applies the product rule but also rewrites the first factor incorrectly, changing (5*x-5/4) to (5*x-5*5/4/5). This is two changes in one step and the rewritten factor is not algebraically equivalent to the original.qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 3 incorrectly applies the product rule by rewriting the first term of the product (5*x - 5/4) as a derivative expression (Derivative(5*x - 5*5/4/5, x)), which is not part of the product rule. The product rule should yield Derivative(5*x - 5/4, x) * log(4*x - 1), not a derivative of a rewritten constant-adjusted term.
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.