Derivative of \( \displaystyle 5 x \ln{\left(5 x + 1 \right)} - 5 x + \ln{\left(5 x + 1 \right)} \)
Problem 2.682 · hard
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(5 x + 1 \right)} - 5 x + \ln{\left(5 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(5 x \ln{\left(5 x + 1 \right)} - 5 x + \ln{\left(5 x + 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \ln{\left(5 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 5 x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \ln{\left(5 x + 1 \right)} - 5 \]constantDifferentiate the linear term 5*x.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 5 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \ln{\left(5 x + 1 \right)} - 5 \]productApply the product rule to the first term.✓ Proved
- \[ = \frac{25 x}{5 x + 1} + 5 \ln{\left(5 x + 1 \right)} - 5 + \frac{5}{5 x + 1} \]chain algebraApply the chain rule to the log terms. Simplify the products.✓ Proved
- \[ = 5 \ln{\left(5 x + 1 \right)} - 5 + \frac{25 x + 5}{5 x + 1} \]algebra algebraCombine the fractions. Factor the numerator.✓ Proved
- \[ = 5 \ln{\left(5 x + 1 \right)} \]simplify simplifyCancel the common factor in the fraction. Final simplification.✓ Proved
Answer \( 5 \log{\left(5 x + 1 \right)} \)
✓ 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 undefined where 5*x + 1 = 0 |
| 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 |
| 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-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.