Derivative of \( \displaystyle \frac{\ln{\left(4 x - 3 \right)}}{\left(4 x - 3\right)^{2}} \)
Problem 2.640 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(4 x - 3 \right)}}{\left(4 x - 3\right)^{2}} \).
- \[ \frac{d}{d x} \frac{\ln{\left(4 x - 3 \right)}}{\left(4 x - 3\right)^{2}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \ln{\left(4 x - 3 \right)} \frac{d}{d x} \frac{1}{\left(4 x - 3\right)^{2}} + \frac{\frac{d}{d x} \ln{\left(4 x - 3 \right)}}{\left(4 x - 3\right)^{2}} \]quotientApply the quotient rule.✓ Proved
- \[ = \ln{\left(4 x - 3 \right)} \frac{d}{d x} \frac{1}{\left(4 x - 3\right)^{2}} + \frac{4}{\left(4 x - 3\right)^{3}} \]chain algebraDifferentiate the numerator using the chain rule. Simplify the first term.✓ Proved
- \[ = - \frac{8 \ln{\left(4 x - 3 \right)}}{\left(4 x - 3\right)^{3}} + \frac{4}{\left(4 x - 3\right)^{3}} \]chain algebra algebraDifferentiate the power term using the chain rule. Rewrite the negative exponent. Distribute the log term.✓ Proved
- \[ = \frac{4 - 8 \ln{\left(4 x - 3 \right)}}{\left(4 x - 3\right)^{3}} \]simplifyCombine the terms over a common denominator.✓ Proved
Answer \( \frac{4 \left(1 - 2 \log{\left(4 x - 3 \right)}\right)}{\left(4 x - 3\right)^{3}} \)
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 undefined where 4*x - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 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 undefined where 4*x - 3 = 0 |
| 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 2 incorrectly applies the quotient rule. The function is a product of log(4*x-3) and (4*x-3)^(-2); the correct rule is the product rule, not the quotient rule. The labeling and the algebraic form are therefore wrong.qwen3.6:27b-mlx: fail (style) — Step 2 applies the quotient rule to a function written as a product (log(...) * (4*x - 3)**(-2)), which is technically incorrect labeling; it should be labeled 'product'. Step 3 applies the chain rule to the numerator but leaves the denominator untouched, which is a valid single-rule application, but the label 'chain' is acceptable for differentiating the composite function log(4x-3). However, the primary defect is Step 2: the input to the derivative operator in Step 1 is a quotient, so applying the quotient rule is correct in spirit, but the expression in Step 2 is the result of the quotient rule applied to u/v where u=log(4x-3) and v=(4x-3)^2. Wait, the quotient rule is (u'v - uv')/v^2. Step 2 shows u'/v + u*v' (which is the product rule form for u*v^-1). The step label says 'quotient' but the algebraic form is that of the product rule (or quotient rule with a specific rearrangement). Actually, the quotient rule yields (u'v - uv')/v^2 = u'/v - uv'/v^2. Step 2 has u'/v + u*v'. This is algebraically incorrect for the quotient rule unless the second term is handled differently. Let's look closer. Step 2: Derivative(log(4*x - 3), x)/(4*x - 3)**2 + log(4*x - 3)*Derivative((4*x - 3)**(-2), x). This is the Product Rule applied to log(4x-3) * (4x-3)^(-2). The label is 'quotient'. This is a mislabeling. The step applies the product rule (implicitly rewriting the denominator as a negative power first, or just applying product rule to the original form interpreted as product). If it applied the quotient rule, it would be (Derivative(log)*denom - log*Derivative(denom))/denom^2. The step shown is Derivative(log)/denom + log*Derivative(denom^-1)? No, it's Derivative(log)/denom + log*Derivative(denom^-2). This is the product rule applied to f*g where g=(4x-3)^-2. The label 'quotient' is wrong. It should be 'product'. Also, Step 3 applies chain rule to the first term's derivative. Step 5 applies chain rule to the second term's derivative. These are fine. The main issue is Step 2's label.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 2 applies the quotient rule to a function written as a product (log(...) * (4*x - 3)**(-2)), which is technically incorrect labeling; it should be labeled 'product'. Step 3 applies the chain rule to the numerator but leaves the denominator untouched, which is a valid single-rule application, but the label 'chain' is acceptable for differentiating the composite function log(4x-3). However, the primary defect is Step 2: the input to the derivative operator in Step 1 is a quotient, so applying the quotient rule is correct in spirit, but the expression in Step 2 is the result of the quotient rule applied to u/v where u=log(4x-3) and v=(4x-3)^2. Wait, the quotient rule is (u'v - uv')/v^2. Step 2 shows u'/v + u*v' (which is the product rule form for u*v^-1). The step label says 'quotient' but the algebraic form is that of the product rule (or quotient rule with a specific rearrangement). Actually, the quotient rule yields (u'v - uv')/v^2 = u'/v - uv'/v^2. Step 2 has u'/v + u*v'. This is algebraically incorrect for the quotient rule unless the second term is handled differently. Let's look closer. Step 2: Derivative(log(4*x - 3), x)/(4*x - 3)**2 + log(4*x - 3)*Derivative((4*x - 3)**(-2), x). This is the Product Rule applied to log(4x-3) * (4x-3)^(-2). The label is 'quotient'. This is a mislabeling. The step applies the product rule (implicitly rewriting the denominator as a negative power first, or just applying product rule to the original form interpreted as product). If it applied the quotient rule, it would be (Derivative(log)*denom - log*Derivative(denom))/denom^2. The step shown is Derivative(log)/denom + log*Derivative(denom^-1)? No, it's Derivative(log)/denom + log*Derivative(denom^-2). This is the product rule applied to f*g where g=(4x-3)^-2. The label 'quotient' is wrong. It should be 'product'. Also, Step 3 applies chain rule to the first term's derivative. Step 5 applies chain rule to the second term's derivative. These are fine. The main issue is Step 2's label.gpt-oss:20b: fail (error) 2026-09-21 — Step 2 incorrectly applies the quotient rule. The function is a product of log(4*x-3) and (4*x-3)^(-2); the correct rule is the product rule, not the quotient rule. The labeling and the algebraic form are therefore wrong.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 2 is labeled "quotient" but the operation performed is the product rule on u·v⁻²; the label should be "product" (or "rewrite" if viewed as a product).
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 2026-09-26 with SymPy 1.14.0.