Derivative of \( \displaystyle 5 x \ln{\left(4 x - 1 \right)} - 5 x - \frac{5 \ln{\left(4 x - 1 \right)}}{4} \)
Problem 2.1450 · hard
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(4 x - 1 \right)} - 5 x - \frac{5 \ln{\left(4 x - 1 \right)}}{4} \).
- \[ \frac{d}{d x} \left(5 x \ln{\left(4 x - 1 \right)} - 5 x - \frac{5 \ln{\left(4 x - 1 \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{5 \ln{\left(4 x - 1 \right)}}{4} \]sumApply the sum rule to separate the terms.✓ Proved
- \[ = \frac{d}{d x} 5 x \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{5 \ln{\left(4 x - 1 \right)}}{4} - 5 \]constantThe derivative of 5*x is 5.✓ Proved
- \[ = \frac{d}{d x} 5 x \ln{\left(4 x - 1 \right)} - \frac{5 \frac{d}{d x} \ln{\left(4 x - 1 \right)}}{4} - 5 \]constant-multipleFactor out the constant 5/4.✓ Proved
- \[ = \frac{d}{d x} 5 x \ln{\left(4 x - 1 \right)} - 5 - \frac{5 \frac{d}{d x} \left(4 x - 1\right)}{4 \left(4 x - 1\right)} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{d}{d x} 5 x \ln{\left(4 x - 1 \right)} - 5 - \frac{5}{4 x - 1} \]derivative algebraThe derivative of 4*x - 1 is 4. Simplify the constant multiplication.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} 5 x - 5 - \frac{5}{4 x - 1} \]productApply the product rule to the first term.✓ Proved
- \[ = \frac{5 x \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} + 5 \ln{\left(4 x - 1 \right)} - 5 - \frac{5}{4 x - 1} \]chainApply the chain rule to the logarithm in the second term.✓ Proved
- \[ = \frac{20 x}{4 x - 1} + 5 \ln{\left(4 x - 1 \right)} - 5 - \frac{5}{4 x - 1} \]derivative algebraThe derivative of 4*x - 1 is 4. Multiply the constants in the second term.✓ Proved
- \[ = 5 \ln{\left(4 x - 1 \right)} - 5 + \frac{20 x - 5}{4 x - 1} \]algebra algebraCombine the fractions with the same denominator. Factor out 5 from the numerator.✓ Proved
- \[ = 5 \ln{\left(4 x - 1 \right)} \]simplify simplifyCancel the common factor (4*x - 1). Combine the remaining constant terms.✓ Proved
Answer \( 5 \ln{\left(4 x - 1 \right)} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 undefined where 4*x - 1 = 0 |
| 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 |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. All labels are appropriate for the operations performed.gpt-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.