Derivative of \( \displaystyle x - \left(x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \)
Problem 2.2389 · hard Video
Differentiate \( \displaystyle f(x) = x - \left(x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(x - \left(x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x - \frac{d}{d x} \left(x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \]sumApply the difference rule.✓ Proved
- \[ = 1 - \frac{d}{d x} \left(x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \]constantThe derivative of x is 1.✓ Proved
- \[ = - \left(x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} - \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(x + \frac{2}{5}\right) + 1 \]productApply the product rule to the second term.✓ Proved
- \[ = - \left(x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} - \left(\frac{d}{d x} \frac{2}{5} + \frac{d}{d x} x\right) \ln{\left(5 x + 2 \right)} + 1 \]sumDistribute the derivative over the sum inside the parentheses.✓ Proved
- \[ = - \left(x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} - \ln{\left(5 x + 2 \right)} + 1 \]constant algebraThe derivative of the constant 2/5 is 0. Simplify the expression inside the parentheses.✓ Proved
- \[ = - \frac{\left(x + \frac{2}{5}\right) \frac{d}{d x} \left(5 x + 2\right)}{5 x + 2} - \ln{\left(5 x + 2 \right)} + 1 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = - \frac{5 \left(x + \frac{2}{5}\right)}{5 x + 2} - \ln{\left(5 x + 2 \right)} + 1 \]derivativeThe derivative of 5*x + 2 is 5.✓ Proved
- \[ = - \ln{\left(5 x + 2 \right)} \]algebra algebra simplify algebra simplifyMultiply the terms in the second part of the expression. Distribute the 5 into the numerator. Simplify the fraction. Distribute the negative sign. Final simplification.✓ Proved
Answer \( - \ln{\left(5 x + 2 \right)} \)
✓ 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. How we verify
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 5*x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 14 | ✓ 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 in a step-by-step manner. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate and the logic is sound.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate and the logic is sound.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.