Derivative of \( \displaystyle - 5 x + \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} \)
Problem 2.2521 · hard Video
Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(- 5 x + \left(5 x + 2\right) \ln{\left(5 x + 2 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(5 x + 2\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(5 x + 2\right) + \frac{d}{d x} \left(- 5 x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = \left(5 x + 2\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(5 x + 2\right) - 5 \]constantDifferentiate the first term.✓ Proved
- \[ = \left(5 x + 2\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + 5 \ln{\left(5 x + 2 \right)} - 5 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = 5 \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(5 x + 2\right) - 5 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = 5 \ln{\left(5 x + 2 \right)} \]derivative algebra algebra simplifyDifferentiate the inner function of the chain rule. Simplify the expression inside the parentheses. Cancel the common factor (5*x + 2). Combine the constant terms.✓ Proved
Answer \( 5 \ln{\left(5 x + 2 \right)} \)
✓ 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 |
| 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 |
| 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 -5*x as "constant"; it should be "derivative". All other steps are correct and apply only one rule each.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.gpt-oss:20b: fail (style) 2026-10-29 — Step 4 incorrectly labels the differentiation of -5*x as "constant"; it should be "derivative". All other steps are correct and apply only one rule each.qwen3.6:27b-mlx: pass 2026-10-28gpt-oss:20b: fail (style) 2026-10-28 — Step 4 labels the differentiation of −5*x as "constant", but the correct rule name is "derivative". No other mathematical errors are present.qwen3.6:27b-mlx: pass 2026-10-28gpt-oss:20b: fail (style) 2026-10-28 — Step 4 incorrectly labels the differentiation of -5*x as "constant"; it should be labeled "derivative". No mathematical errors otherwise.
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 2026-10-29 with SymPy 1.14.0.