Derivative of \( \displaystyle - 5 x + \left(5 x - 3\right) \ln{\left(5 x - 3 \right)} \)
Problem 2.2396 · hard Video
Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x - 3\right) \ln{\left(5 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(- 5 x + \left(5 x - 3\right) \ln{\left(5 x - 3 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x - 3\right) \ln{\left(5 x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(5 x - 3\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right) + \frac{d}{d x} \left(- 5 x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = \left(5 x - 3\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right) - 5 \]constantDifferentiate the constant term -5.✓ Proved
- \[ = \left(5 x - 3\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + 5 \ln{\left(5 x - 3 \right)} - 5 \]derivativeDifferentiate the first part of the product term.✓ Proved
- \[ = 5 \ln{\left(5 x - 3 \right)} + \frac{d}{d x} \left(5 x - 3\right) - 5 \]logarithmicApply the derivative rule for the logarithm.✓ Proved
- \[ = 5 \ln{\left(5 x - 3 \right)} \]derivative algebra simplifyDifferentiate the inner function of the chain rule. Simplify the product of the terms. Combine like terms.✓ Proved
Answer \( 5 \ln{\left(5 x - 3 \right)} \)
✓ 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 |
| 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 labeled "derivative". No other errors are present.qwen3.6:27b-mlx: pass — The solution correctly applies the sum, product, chain, and logarithmic derivative rules in separate steps. The labels accurately reflect the operations performed, and the algebraic simplifications are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the sum, product, chain, and logarithmic derivative rules in separate steps. The labels accurately reflect the operations performed, and the algebraic simplifications are correct.gpt-oss:20b: fail (style) 2026-10-18 — Step 4 incorrectly labels the differentiation of -5*x as "constant"; it should be labeled "derivative". No other errors are present.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules step-by-step. Each step changes only one aspect of the expression, and the labels accurately reflect the rules applied (sum, product, chain/logarithmic, algebraic simplification).gpt-oss:20b: fail (style) 2026-10-16 — Step 4 incorrectly labels the differentiation of “-5*x” as a "constant" rule; it should be a "derivative" (or "algebra" after the derivative is taken).qwen3.6:27b-mlx: pass 2026-10-15gpt-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.