Derivative of \( \displaystyle 3 x \ln{\left(5 x + 2 \right)} - 3 x + \frac{6 \ln{\left(5 x + 2 \right)}}{5} \)
Problem 2.1220 · hard
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(5 x + 2 \right)} - 3 x + \frac{6 \ln{\left(5 x + 2 \right)}}{5} \).
- \[ \frac{d}{d x} \left(3 x \ln{\left(5 x + 2 \right)} - 3 x + \frac{6 \ln{\left(5 x + 2 \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{6 \ln{\left(5 x + 2 \right)}}{5} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{6 \ln{\left(5 x + 2 \right)}}{5} - 3 \]constantDifferentiate the linear term 3*x.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(5 x + 2 \right)} + 3 \ln{\left(5 x + 2 \right)} + \frac{6 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} - 3 \]productApply the product rule to the first term.✓ Proved
- \[ = \frac{3 x \frac{d}{d x} \left(5 x + 2\right)}{5 x + 2} + 3 \ln{\left(5 x + 2 \right)} + \frac{6 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} - 3 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{15 x}{5 x + 2} + 3 \ln{\left(5 x + 2 \right)} + \frac{6 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} - 3 \]derivative algebra algebraDifferentiate the inner function 5*x + 2. Simplify the fraction. Multiply the numerator.✓ Proved
- \[ = \frac{15 x}{5 x + 2} + 3 \ln{\left(5 x + 2 \right)} - 3 + \frac{6}{5 x + 2} \]chain derivative algebraApply the chain rule to the last term. Differentiate the logarithm in the last term. Simplify the constant multiple.✓ Proved
- \[ = 3 \ln{\left(5 x + 2 \right)} - 3 + \frac{15 x + 6}{5 x + 2} \]algebraCombine the fractions.✓ Proved
- \[ = 3 \ln{\left(5 x + 2 \right)} \]algebra algebra algebra simplifyFind a common denominator for the constant term. Distribute the -3. Simplify the numerator. Final simplification.✓ Proved
Answer \( 3 \ln{\left(5 x + 2 \right)} \)
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 undefined where 5*x + 2 = 0 |
| 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 undefined where 5*x + 2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 16 | ✓ 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 step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the algebraic simplification is sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — 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, and the algebraic simplification is sound.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 applies the derivative rule to the term 3*x but incorrectly labels it 'constant'. The derivative of 3*x is 3, which requires the power rule or derivative rule, not the constant rule (which applies to terms with no x). Additionally, Step 3 only differentiates one term of the sum, violating the 'one thing per step' constraint if interpreted as a full simplification, but the primary defect is the incorrect label for differentiating a variable term.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.