Derivative of \( \displaystyle 2 x \ln{\left(5 x + 2 \right)} - 2 x + \frac{4 \ln{\left(5 x + 2 \right)}}{5} \)
Problem 2.1446 · hard
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(5 x + 2 \right)} - 2 x + \frac{4 \ln{\left(5 x + 2 \right)}}{5} \).
- \[ \frac{d}{d x} \left(2 x \ln{\left(5 x + 2 \right)} - 2 x + \frac{4 \ln{\left(5 x + 2 \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{4 \ln{\left(5 x + 2 \right)}}{5} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 2 x \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{4 \ln{\left(5 x + 2 \right)}}{5} - 2 \]constantThe derivative of 2*x is 2.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \frac{4 \ln{\left(5 x + 2 \right)}}{5} - 2 \]productApply the product rule to the first term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(5 x + 2 \right)} + 2 \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{4 \ln{\left(5 x + 2 \right)}}{5} - 2 \]constant-multipleThe derivative of 2*x is 2.✓ Proved
- \[ = \frac{10 x}{5 x + 2} + 2 \ln{\left(5 x + 2 \right)} + \frac{4 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} - 2 \]chainApply the chain rule to the log terms.✓ Proved
- \[ = \frac{10 x}{5 x + 2} + 2 \ln{\left(5 x + 2 \right)} - 2 + \frac{4}{5 x + 2} \]algebra algebraSimplify the derivative of the log term. Simplify the product term.✓ Proved
- \[ = 2 \ln{\left(5 x + 2 \right)} - 2 + \frac{10 x + 4}{5 x + 2} \]algebraCombine the fractions.✓ Proved
- \[ = 2 \ln{\left(5 x + 2 \right)} \]algebra algebra algebra simplifyFind a common denominator for the constant term. Distribute the -2. Simplify the numerator. Final simplification.✓ Proved
Answer \( 2 \ln{\left(5 x + 2 \right)} \)
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 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 undefined where 5*x + 2 = 0 |
| 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: passqwen3.6:27b-mlx: fail (error) — Step 5 is labeled 'constant-multiple' but performs two distinct operations: it evaluates the derivative of 2*x (a 'derivative' step) and pulls the constant 4/5 out of the derivative operator (a 'constant-multiple' step). This violates the one-rule-per-step constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 is labeled 'constant-multiple' but performs two distinct operations: it evaluates the derivative of 2*x (a 'derivative' step) and pulls the constant 4/5 out of the derivative operator (a 'constant-multiple' step). This violates the one-rule-per-step constraint.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.