Derivative of \( \displaystyle - x + \left(x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \)
Problem 2.1924 · hard Beautiful
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
- \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(x + \frac{2}{5}\right) + \frac{d}{d x} \left(- x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = \left(x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(x + \frac{2}{5}\right) - 1 \]constantDifferentiate the first term.✓ Proved
- \[ = \left(x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} - 1 \]derivative constantDifferentiate the sum inside the first derivative term. Differentiate the constant 2/5.✓ 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
- \[ = \frac{5 \left(x + \frac{2}{5}\right)}{5 x + 2} + \ln{\left(5 x + 2 \right)} - 1 \]derivativeDifferentiate the inner function 5*x + 2.✓ Proved
- \[ = \ln{\left(5 x + 2 \right)} \]algebra algebra algebra algebra simplifySimplify the expression. Simplify the product of the constant and the fraction. Distribute the 5 into the numerator. Simplify the fraction. Combine the constant terms.✓ Proved
Answer \( \ln{\left(5 x + 2 \right)} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 + 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 |
| 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 |
| 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 (error) — Step 9 applies multiple simplifications at once (cancelling fractions, multiplying constants, and simplifying the product), violating the rule that each step must change only one thing. This makes the step invalid under the given labeling conventions.qwen3.6:27b-mlx: fail (error) — Step 4 is labeled 'constant' but performs the differentiation of -x to -1, which is a 'derivative' step. Step 5 is labeled 'derivative' but performs the differentiation of the sum x + 2/5 into 1 + 0, which involves the 'sum' rule and 'derivative' rule, violating the one-rule-per-step constraint. Step 6 is labeled 'constant' but simplifies 1+0 to 1, which is 'simplify' or 'algebra', not a differentiation rule.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Step 9 applies multiple simplifications at once (cancelling fractions, multiplying constants, and simplifying the product), violating the rule that each step must change only one thing. This makes the step invalid under the given labeling conventions.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 is labeled 'constant' but performs the differentiation of -x to -1, which is a 'derivative' step. Step 5 is labeled 'derivative' but performs the differentiation of the sum x + 2/5 into 1 + 0, which involves the 'sum' rule and 'derivative' rule, violating the one-rule-per-step constraint. Step 6 is labeled 'constant' but simplifies 1+0 to 1, which is 'simplify' or 'algebra', not a differentiation rule.gpt-oss:20b: fail (error) 2026-10-09 — Step 9 applies several algebraic simplifications in one move (combining fractions, canceling terms, and introducing extra factors), violating the rule that each step must change only one thing. This makes the step non‑granular and incorrect under the contract.qwen3.6:27b-mlx: pass 2026-10-09 — 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 2026-10-09 with SymPy 1.14.0.