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 Video
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. 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 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: fail (style) — Step 10 incorrectly labels the operation as "derivative"; it is merely an algebraic simplification of the fraction, not a differentiation step.qwen3.6:27b-mlx: fail (style) — Step 3 applies the derivative rule to the term 3*x but labels it 'constant', which is incorrect as 3*x is not a constant; it should be labeled 'derivative' or 'constant-multiple'. Additionally, Step 10 labels the differentiation of the inner function as 'derivative' but the step actually simplifies the result of the chain rule application from Step 9, making the label confusing or redundant depending on interpretation, though Step 3 is a clear mislabeling.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 3 applies the derivative rule to the term 3*x but labels it 'constant', which is incorrect as 3*x is not a constant; it should be labeled 'derivative' or 'constant-multiple'. Additionally, Step 10 labels the differentiation of the inner function as 'derivative' but the step actually simplifies the result of the chain rule application from Step 9, making the label confusing or redundant depending on interpretation, though Step 3 is a clear mislabeling.gpt-oss:20b: fail (style) 2026-10-03 — Step 10 incorrectly labels the operation as "derivative"; it is merely an algebraic simplification of the fraction, not a differentiation step.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 2026-10-03 with SymPy 1.14.0.