Derivative of \( \displaystyle x \ln{\left(5 x + 1 \right)} - x + \frac{\ln{\left(5 x + 1 \right)}}{5} \)
Problem 2.648 · hard Beautiful
Differentiate \( \displaystyle f(x) = x \ln{\left(5 x + 1 \right)} - x + \frac{\ln{\left(5 x + 1 \right)}}{5} \).
- \[ \frac{d}{d x} \left(x \ln{\left(5 x + 1 \right)} - x + \frac{\ln{\left(5 x + 1 \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} - 1 \]constantThe derivative of x is 1.✓ Proved
- \[ = x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} - 1 \]productApply the product rule to the first term.✓ Proved
- \[ = x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} - 1 \]constantThe derivative of x is 1.✓ Proved
- \[ = \frac{x \frac{d}{d x} \left(5 x + 1\right)}{5 x + 1} + \ln{\left(5 x + 1 \right)} + \frac{\frac{d}{d x} \ln{\left(5 x + 1 \right)}}{5} - 1 \]chainApply the chain rule to the log terms.✓ Proved
- \[ = \frac{5 x}{5 x + 1} + \ln{\left(5 x + 1 \right)} - 1 + \frac{1}{5 x + 1} \]derivative algebraDifferentiate the inner function 5x + 1. Simplify the terms.✓ Proved
- \[ = \ln{\left(5 x + 1 \right)} \]algebra algebra simplifyCombine the fractions. Simplify the fraction. Final simplification.✓ Proved
Answer \( \log{\left(5 x + 1 \right)} \)
✓ 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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ 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) — Steps 3 and 5 incorrectly label the derivative of x as "constant"; the correct rule name is "derivative". No mathematical errors otherwise.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Steps 3 and 5 incorrectly label the derivative of x as "constant"; the correct rule name is "derivative". No mathematical errors otherwise.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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 2026-09-26 with SymPy 1.14.0.