Derivative of \( \displaystyle 2 x \ln{\left(5 x + 1 \right)} - 2 x + \frac{2 \ln{\left(5 x + 1 \right)}}{5} \)
Problem 2.1393 · hard
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(5 x + 1 \right)} - 2 x + \frac{2 \ln{\left(5 x + 1 \right)}}{5} \).
- \[ \frac{d}{d x} \left(2 x \ln{\left(5 x + 1 \right)} - 2 x + \frac{2 \ln{\left(5 x + 1 \right)}}{5}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{2 \ln{\left(5 x + 1 \right)}}{5} \]sum constant-multipleApply the sum rule. Factor out the constant from the third term.✓ Proved
- \[ = \frac{d}{d x} 2 x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{2 \ln{\left(5 x + 1 \right)}}{5} - 2 \]derivativeDifferentiate the second term.✓ Proved
- \[ = 2 \frac{d}{d x} x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{2 \ln{\left(5 x + 1 \right)}}{5} - 2 \]constant-multipleFactor out the constant from the first term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 2 \ln{\left(5 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{2 \ln{\left(5 x + 1 \right)}}{5} - 2 \]productApply the product rule to the first term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 2 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{2 \ln{\left(5 x + 1 \right)}}{5} - 2 \]derivativeDifferentiate x.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 2 \ln{\left(5 x + 1 \right)} + \frac{2 \frac{d}{d x} \ln{\left(5 x + 1 \right)}}{5} - 2 \]algebraDistribute the 2.✓ Proved
- \[ = \left(2 x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 2 \ln{\left(5 x + 1 \right)} - 2 \]algebraGroup the derivative terms.✓ Proved
- \[ = \frac{\left(2 x + \frac{2}{5}\right) \frac{d}{d x} \left(5 x + 1\right)}{5 x + 1} + 2 \ln{\left(5 x + 1 \right)} - 2 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \left(2 x + \frac{2}{5}\right)}{5 x + 1} + 2 \ln{\left(5 x + 1 \right)} - 2 \]derivativeDifferentiate the inner function 5*x + 1.✓ Proved
- \[ = 2 \ln{\left(5 x + 1 \right)} - 2 + \frac{10 x + 2}{5 x + 1} \]algebraMultiply the numerator.✓ Proved
- \[ = 2 \ln{\left(5 x + 1 \right)} \]algebra algebra algebra simplifyFind a common denominator. Distribute the -2. Combine like terms in the numerator. Simplify the expression.✓ Proved
Answer \( 2 \ln{\left(5 x + 1 \right)} \)
✓ 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 |
| 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.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. The algebraic simplification in the final steps is valid and clearly labeled.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.