Derivative of \( \displaystyle x + \left(\frac{1}{5} - x\right) \ln{\left(5 x - 1 \right)} \)
Problem 2.1986 · hard
Differentiate \( \displaystyle f(x) = x + \left(\frac{1}{5} - x\right) \ln{\left(5 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(x + \left(\frac{1}{5} - x\right) \ln{\left(5 x - 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \left(\frac{1}{5} - x\right) \ln{\left(5 x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(\frac{1}{5} - x\right) \ln{\left(5 x - 1 \right)} + 1 \]constantThe derivative of x is 1.✓ Proved
- \[ = \left(\frac{1}{5} - x\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} + \ln{\left(5 x - 1 \right)} \frac{d}{d x} \left(\frac{1}{5} - x\right) + 1 \]productApply the product rule to the second term.✓ Proved
- \[ = \left(\frac{1}{5} - x\right) \frac{d}{d x} \ln{\left(5 x - 1 \right)} - \ln{\left(5 x - 1 \right)} + 1 \]constant algebraThe derivative of 1/5 - x is -1. Distribute the negative sign.✓ Proved
- \[ = \frac{\left(\frac{1}{5} - x\right) \frac{d}{d x} \left(5 x - 1\right)}{5 x - 1} - \ln{\left(5 x - 1 \right)} + 1 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \left(\frac{1}{5} - x\right)}{5 x - 1} - \ln{\left(5 x - 1 \right)} + 1 \]constantThe derivative of 5*x - 1 is 5.✓ Proved
- \[ = \frac{1 - 5 x}{5 x - 1} - \ln{\left(5 x - 1 \right)} + 1 \]algebra algebra algebra algebraMultiply the terms together. Simplify the numerator. Rearrange the numerator. Factor out -1 from the numerator.✓ Proved
- \[ = - \ln{\left(5 x - 1 \right)} \]simplify simplifyCancel the common term in the fraction. Combine the constant terms.✓ Proved
Answer \( - \ln{\left(5 x - 1 \right)} \)
✓ 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 undefined where 5*x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 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 |
| 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 valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — 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 valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
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 2026-10-10 with SymPy 1.14.0.