Derivative of \( \displaystyle - 3 x + \left(3 x - \frac{9}{5}\right) \ln{\left(5 x - 3 \right)} \)
Problem 2.1929 · hard
Differentiate \( \displaystyle f(x) = - 3 x + \left(3 x - \frac{9}{5}\right) \ln{\left(5 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(- 3 x + \left(3 x - \frac{9}{5}\right) \ln{\left(5 x - 3 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x - \frac{9}{5}\right) \ln{\left(5 x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(3 x - \frac{9}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} \frac{d}{d x} \left(3 x - \frac{9}{5}\right) + \frac{d}{d x} \left(- 3 x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = - \left(3 x - \frac{9}{5}\right) \frac{d}{d x} 1 + \left(3 x - \frac{9}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 3 x\right) \]sumDistribute the derivative over the subtraction.✓ Proved
- \[ = - \left(3 x - \frac{9}{5}\right) \frac{d}{d x} \frac{9}{5} + \left(3 x - \frac{9}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 3 x\right) \]algebraSimplify the constant term.✓ Proved
- \[ = - \left(3 x - \frac{9}{5}\right) \frac{d}{d x} \frac{9}{5} + \left(3 x - \frac{9}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + 3 \ln{\left(5 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) \]derivativeDifferentiate the linear term 3*x.✓ Proved
- \[ = \left(3 x - \frac{9}{5}\right) \frac{d}{d x} \ln{\left(5 x - 3 \right)} + 3 \ln{\left(5 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) \]derivative algebraDifferentiate the constant term 9/5. The term with zero vanishes.✓ Proved
- \[ = \frac{\left(3 x - \frac{9}{5}\right) \frac{d}{d x} \left(5 x - 3\right)}{5 x - 3} + 3 \ln{\left(5 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \left(3 x - \frac{9}{5}\right)}{5 x - 3} + 3 \ln{\left(5 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) \]derivativeDifferentiate the inner function 5*x - 3.✓ Proved
- \[ = 3 \ln{\left(5 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{15 x - 9}{5 x - 3} \]algebraMultiply the constant 5 into the numerator.✓ Proved
- \[ = 3 \ln{\left(5 x - 3 \right)} - 3 + \frac{15 x - 9}{5 x - 3} \]derivative algebraDifferentiate the remaining -3*x term. Factor the numerator.✓ Proved
- \[ = 3 \ln{\left(5 x - 3 \right)} \]simplify simplifyCancel the common factor (5*x - 3). Combine the constant terms.✓ Proved
Answer \( 3 \ln{\left(5 x - 3 \right)} \)
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 - 3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 3 = 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 4 incorrectly rewrites the derivative of (3*x-9/5) as a sum involving 5/5 and 9/5, applying two rules at once and producing an algebraic error. The correct derivative is simply 3, and the step should not introduce extra terms.qwen3.6:27b-mlx: fail (style) — Step 4 splits the derivative of a difference into two separate derivative terms, which is an application of the sum/difference rule, but it also appears to rewrite the constant 9/5 as 5/5 in the note or implicitly, violating the one-change-per-step rule if combined with the split. More critically, Step 4's note says 'Distribute the derivative over the subtraction', but the step actually applies the linearity of the derivative operator (sum rule) to split `Derivative(A - B)` into `Derivative(A) - Derivative(B)`. While 'sum' is the correct label for linearity, the step also seems to prepare for the next step by isolating the constant term. However, the primary defect is in Step 4 where the term `Derivative(3*x - 9/5, x)` is split into `Derivative(3*x, x) - Derivative(9/5, x)`. This is a valid application of the sum rule (linearity). But look at Step 4 again: it changes `Derivative(3*x - 9/5, x)` to `Derivative(3*x, x) - (3*x - 9/5)*Derivative(5/5, x)`? No, it changes it to `Derivative(3*x, x)*log(...) - (3*x - 9/5)*Derivative(5/5, x)`. Wait, Step 3 has `Derivative(3*x - 9/5, x)*log(5*x - 3)`. Step 4 changes this to `Derivative(3*x, x)*log(5*x - 3) - (3*x - 9/5)*Derivative(5/5, x)`. This step applies the sum rule to the derivative `Derivative(3*x - 9/5, x)` AND rewrites `9/5` as `5/5`? No, it rewrites `Derivative(9/5, x)` as `Derivative(5/5, x)`? Actually, Step 4 introduces `Derivative(5/5, x)` which is weird. Step 5 changes `5/5` to `9/5`. This is just algebraic rewriting of a constant inside a derivative operator, which is unnecessary and confusing. The main issue is that Step 4 applies the sum rule to split the derivative, but the resulting term `Derivative(5/5, x)` is an odd intermediate form. Let's look closer. Step 4: `Derivative(3*x - 9/5, x)` becomes `Derivative(3*x, x) - Derivative(9/5, x)` effectively, but written as `Derivative(3*x, x) - (3*x - 9/5)*Derivative(5/5, x)`? No, the term is `Derivative(3*x - 9/5, x)*log(...)`. The step changes it to `Derivative(3*x, x)*log(...) - (3*x - 9/5)*Derivative(5/5, x)`. This is not a standard application of the sum rule. The sum rule would give `Derivative(3*x, x)*log(...) + Derivative(-9/5, x)*log(...)`. The step instead seems to have distributed the log term incorrectly or mixed rules. Actually, looking at Step 3: `Derivative(3*x - 9/5, x)*log(5*x - 3)`. Step 4: `Derivative(3*x, x)*log(5*x - 3) - (3*x - 9/5)*Derivative(5/5, x)`. This step is mathematically nonsensical as a single rule application. It looks like it tried to apply the product rule again or something else. It's a mess. It violates the one-rule-per-step constraint by mixing linearity with strange algebraic substitutions.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Step 4 incorrectly rewrites the derivative of (3*x-9/5) as a sum involving 5/5 and 9/5, applying two rules at once and producing an algebraic error. The correct derivative is simply 3, and the step should not introduce extra terms.qwen3.6:27b-mlx: fail (style) 2026-10-09 — Step 4 splits the derivative of a difference into two separate derivative terms, which is an application of the sum/difference rule, but it also appears to rewrite the constant 9/5 as 5/5 in the note or implicitly, violating the one-change-per-step rule if combined with the split. More critically, Step 4's note says 'Distribute the derivative over the subtraction', but the step actually applies the linearity of the derivative operator (sum rule) to split `Derivative(A - B)` into `Derivative(A) - Derivative(B)`. While 'sum' is the correct label for linearity, the step also seems to prepare for the next step by isolating the constant term. However, the primary defect is in Step 4 where the term `Derivative(3*x - 9/5, x)` is split into `Derivative(3*x, x) - Derivative(9/5, x)`. This is a valid application of the sum rule (linearity). But look at Step 4 again: it changes `Derivative(3*x - 9/5, x)` to `Derivative(3*x, x) - (3*x - 9/5)*Derivative(5/5, x)`? No, it changes it to `Derivative(3*x, x)*log(...) - (3*x - 9/5)*Derivative(5/5, x)`. Wait, Step 3 has `Derivative(3*x - 9/5, x)*log(5*x - 3)`. Step 4 changes this to `Derivative(3*x, x)*log(5*x - 3) - (3*x - 9/5)*Derivative(5/5, x)`. This step applies the sum rule to the derivative `Derivative(3*x - 9/5, x)` AND rewrites `9/5` as `5/5`? No, it rewrites `Derivative(9/5, x)` as `Derivative(5/5, x)`? Actually, Step 4 introduces `Derivative(5/5, x)` which is weird. Step 5 changes `5/5` to `9/5`. This is just algebraic rewriting of a constant inside a derivative operator, which is unnecessary and confusing. The main issue is that Step 4 applies the sum rule to split the derivative, but the resulting term `Derivative(5/5, x)` is an odd intermediate form. Let's look closer. Step 4: `Derivative(3*x - 9/5, x)` becomes `Derivative(3*x, x) - Derivative(9/5, x)` effectively, but written as `Derivative(3*x, x) - (3*x - 9/5)*Derivative(5/5, x)`? No, the term is `Derivative(3*x - 9/5, x)*log(...)`. The step changes it to `Derivative(3*x, x)*log(...) - (3*x - 9/5)*Derivative(5/5, x)`. This is not a standard application of the sum rule. The sum rule would give `Derivative(3*x, x)*log(...) + Derivative(-9/5, x)*log(...)`. The step instead seems to have distributed the log term incorrectly or mixed rules. Actually, looking at Step 3: `Derivative(3*x - 9/5, x)*log(5*x - 3)`. Step 4: `Derivative(3*x, x)*log(5*x - 3) - (3*x - 9/5)*Derivative(5/5, x)`. This step is mathematically nonsensical as a single rule application. It looks like it tried to apply the product rule again or something else. It's a mess. It violates the one-rule-per-step constraint by mixing linearity with strange algebraic substitutions.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 incorrectly applies the sum rule to split the derivative of a product term, introducing a spurious term involving the derivative of a constant (9/5) that does not exist in the original expression. This is a fundamental logical error in the differentiation process.gpt-oss:20b: fail (error) 2026-10-09 — Step 4 incorrectly applies the product rule: it splits the derivative of (3*x-9/5) into two separate derivatives, one of which is nonsensical (Derivative(5/5,x)). This applies more than one rule at once and introduces a term that does not exist. The subsequent steps propagate this mistake, leading to an incorrect final result.
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.