Derivative of \( \displaystyle - 2 x + \left(2 x + \frac{4}{5}\right) \ln{\left(5 x + 2 \right)} \)
Problem 2.2135 · hard
Differentiate \( \displaystyle f(x) = - 2 x + \left(2 x + \frac{4}{5}\right) \ln{\left(5 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(- 2 x + \left(2 x + \frac{4}{5}\right) \ln{\left(5 x + 2 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(2 x + \frac{4}{5}\right) \ln{\left(5 x + 2 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(2 x + \frac{4}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \left(- 2 x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = \left(2 x + \frac{4}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + 2 \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) \]constant-multipleDifferentiate the linear term 2*x.✓ Proved
- \[ = \frac{\left(2 x + \frac{4}{5}\right) \frac{d}{d x} \left(5 x + 2\right)}{5 x + 2} + 2 \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \left(2 x + \frac{4}{5}\right)}{5 x + 2} + 2 \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) \]derivativeDifferentiate the inner function 5*x + 2.✓ Proved
- \[ = 2 \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) + \frac{10 x + 4}{5 x + 2} \]algebra algebraMultiply the constant 5 into the numerator. Factor out a 2 from the numerator.✓ Proved
- \[ = 2 \ln{\left(5 x + 2 \right)} \]derivative simplifyDifferentiate the first term -2*x. Simplify the expression by combining constants.✓ Proved
Answer \( 2 \ln{\left(5 x + 2 \right)} \)
✓ 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.
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 + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 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 |
| 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) — Step 4 incorrectly labels the differentiation of 2*x as a constant‑multiple rule; it should be a derivative step. All other steps are correct.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is correct.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-12 — Step 4 incorrectly labels the differentiation of 2*x as a constant‑multiple rule; it should be a derivative step. All other steps are correct.qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is correct.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is correct.
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 2026-10-12 with SymPy 1.14.0.