Derivative of \( \displaystyle - 2 x + \left(2 x + \frac{2}{5}\right) \ln{\left(5 x + 1 \right)} \)
Problem 2.2125 · hard
Differentiate \( \displaystyle f(x) = - 2 x + \left(2 x + \frac{2}{5}\right) \ln{\left(5 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 2 x + \left(2 x + \frac{2}{5}\right) \ln{\left(5 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(2 x + \frac{2}{5}\right) \ln{\left(5 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} 2 x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{2 \ln{\left(5 x + 1 \right)}}{5} \]productApply the product rule to the second term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \frac{2 \ln{\left(5 x + 1 \right)}}{5} \]productApply the product rule to the term 2*x*log(5*x + 1).✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 2 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \frac{2 \ln{\left(5 x + 1 \right)}}{5} \]derivativeDifferentiate 2*x.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 2 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 2 x\right) + \frac{2 \frac{d}{d x} \ln{\left(5 x + 1 \right)}}{5} \]constant-multiple constantApply the constant multiple rule to the third term. The derivative of a constant is zero.✓ Proved
- \[ = \left(2 x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} + 2 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 2 x\right) \]algebraFactor out the common derivative term.✓ Proved
- \[ = \frac{\left(2 x + \frac{2}{5}\right) \frac{d}{d x} \left(5 x + 1\right)}{5 x + 1} + 2 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 2 x\right) \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \left(2 x + \frac{2}{5}\right)}{5 x + 1} + 2 \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- 2 x\right) \]derivativeDifferentiate the inner function 5*x + 1.✓ Proved
- \[ = \frac{5 \left(2 x + \frac{2}{5}\right)}{5 x + 1} + 2 \ln{\left(5 x + 1 \right)} - 2 \]derivativeDifferentiate -2*x.✓ Proved
- \[ = 2 \ln{\left(5 x + 1 \right)} - 2 + \frac{10 x + 2}{5 x + 1} \]algebra algebraMultiply the fraction by 5. Factor the numerator.✓ Proved
- \[ = 2 \ln{\left(5 x + 1 \right)} \]simplify simplifySimplify the fraction. Combine the constant terms.✓ Proved
Answer \( 2 \ln{\left(5 x + 1 \right)} \)
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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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 undefined where 5*x + 1 = 0 |
| 15 | ✓ 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 3 incorrectly applies the product rule to split the term (2*x + 2/5)*log(5*x + 1) into 2*x*log(5*x + 1) + (2/5)*log(5*x + 1). This is a distributive expansion, not a product rule application, and thus violates the single‑rule‑per‑step rule. Additionally, step 7 adds a zero term that is not justified by any rule, further indicating a mistake in the derivation sequence.qwen3.6:27b-mlx: fail (error) — Step 3 applies the distributive property (algebra) to split the product term, but labels it 'product'. The product rule is not applied until Step 4. This violates the single-rule-per-step constraint and mislabels the operation.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — Step 3 incorrectly applies the product rule to split the term (2*x + 2/5)*log(5*x + 1) into 2*x*log(5*x + 1) + (2/5)*log(5*x + 1). This is a distributive expansion, not a product rule application, and thus violates the single‑rule‑per‑step rule. Additionally, step 7 adds a zero term that is not justified by any rule, further indicating a mistake in the derivation sequence.qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 3 applies the distributive property (algebra) to split the product term, but labels it 'product'. The product rule is not applied until Step 4. This violates the single-rule-per-step constraint and mislabels the operation.gpt-oss:20b: fail (style) 2026-10-12 — Step 3 incorrectly labels the distributive expansion of (2*x+2/5)*log(5*x+1) as a product rule. It should be an algebraic rewrite, not a product rule application. This mislabeling could mislead a student about when the product rule applies.qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 3 incorrectly labels the expansion of a product as the 'product' rule; it is actually the 'sum' rule (linearity of differentiation) applied to the expanded terms. The product rule is applied in Step 4.
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 2026-10-12 with SymPy 1.14.0.