Derivative of \( \displaystyle \ln{\left(\frac{3 x}{\sqrt{1 - 9 x^{2}}} \right)} \)
Problem 2.1144 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{3 x}{\sqrt{1 - 9 x^{2}}} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\frac{3 x}{\sqrt{1 - 9 x^{2}}} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(\ln{\left(3 x \right)} - \ln{\left(\sqrt{1 - 9 x^{2}} \right)}\right) \]simplify simplifyUse logarithm properties to split the quotient. Rewrite the square root as an exponent.✓ Proved
- \[ = \frac{d}{d x} \left(\ln{\left(3 x \right)} - \frac{\ln{\left(1 - 9 x^{2} \right)}}{2}\right) \]constant-multipleBring the exponent to the front.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(3 x \right)} - \frac{d}{d x} \frac{\ln{\left(1 - 9 x^{2} \right)}}{2} \]sumApply the difference rule.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(3 x \right)} - \frac{\frac{d}{d x} \ln{\left(1 - 9 x^{2} \right)}}{2} \]constant-multipleFactor out the constant.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(3 x \right)} - \frac{\frac{d}{d x} \left(1 - 9 x^{2}\right)}{2 \left(1 - 9 x^{2}\right)} \]chainApply the chain rule to the second term.✓ Proved
- \[ = \frac{9 x}{1 - 9 x^{2}} + \frac{d}{d x} \ln{\left(3 x \right)} \]derivativeDifferentiate the inner function.✓ Proved
- \[ = \frac{9 x}{1 - 9 x^{2}} + \frac{1}{x} \]derivative simplifyDifferentiate the first term using the chain rule. Simplify the second term.✓ Proved
- \[ = \frac{1}{x \left(1 - 9 x^{2}\right)} \]algebra simplifyFind a common denominator. Simplify the numerator.✓ Proved
Answer \( \frac{1}{- 9 x^{3} + x} \)
Mind the domain. The answer is also defined on (-oo, -1/3) and (-1/3, 0) and (1/3, oo), where f(x) is not. Substituting there gives a number that is not a slope of f.
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 1 - 9*x**2 = 0 |
| 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 undefined where 1 - 9*x**2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 1 - 9*x**2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 1 - 9*x**2 = 0 undefined where x = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 - 9*x**2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 - 9*x**2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 - 9*x**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where -9*x**3 + x = 0 |
| 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: fail (style) — [domain objection, downgraded to style] Step 9 claims to differentiate the first term using the chain rule, but the result 1/x is incorrect for d/dx(log(3x)) which should be 3/(3x) = 1/x only if the constant factor is handled, but the label 'derivative' is used for a step that actually performs differentiation of a composite function log(3x) -> 1/(3x)*3 = 1/x. While the result is correct, the step skips the explicit chain rule application for the first term or mislabels the simplification. More critically, Step 9 applies differentiation to the first term while the second term was already differentiated in Step 8. The transition from Step 8 to Step 9 differentiates the first term `Derivative(log(3*x), x)` into `1/x`. This is a differentiation step. However, the note says 'using the chain rule'. The derivative of log(3x) is indeed 1/x. But look at Step 9's label: 'derivative'. This is acceptable. Let's look closer at Step 2. Step 2 splits log(A/B) into log(A) - log(B). This is a logarithmic property. The label is 'simplify'. This is acceptable. Step 4 brings the exponent 1/2 out. Label 'constant-multiple'. Acceptable. Step 7 applies chain rule to log(1-9x^2). Label 'chain'. Acceptable. Step 8 differentiates 1-9x^2. Label 'derivative'. Acceptable. Step 9 differentiates log(3x). The label is 'derivative'. The note says 'using the chain rule'. Is this a defect? The rule 'derivative' covers unfolding d/dx on known forms. log(3x) is not a 'known form' in the base list (x, constant, sin, exp). It requires chain rule or log properties. If the system considers log(kx) a known form, it's fine. But typically, one would use 'chain' or 'logarithmic' (not in list) or 'simplify' (log(3x)=log3+logx). The step jumps from Derivative(log(3x), x) to 1/x. This is a differentiation step. The label 'derivative' is in the vocabulary. Is it wrong? The prompt says 'derivative' means unfolding d/dx on a known form. Is log(3x) a known form? Usually, known forms are x, c, sin(x), exp(x). log(x) is not explicitly listed in the 'derivative' definition text provided in the prompt ('derivative of x, of a constant, of sin, of exp'). It does not list log. Therefore, applying 'derivative' to log(3x) is likely a defect because 'logarithmic' is not in the vocabulary, and 'chain' is not applied explicitly in the label (though mentioned in note). Wait, the vocabulary includes 'logarithmic'. The prompt says: 'Each step names its rule from this fixed vocabulary... logarithmic'. Step 9 uses 'derivative' but the operation is differentiating a logarithm. The rule for differentiating log is 'logarithmic' or 'chain' if composite. Since 'logarithmic' is in the vocabulary, using 'derivative' for a log function is incorrect labeling. The step should likely be 'chain' (if viewing as log(u)) or 'logarithmic' (if viewing as log rule). Given 'logarithmic' is in the list, failing to use it for a log derivative is a labeling error. Also, Step 9 combines the differentiation of the first term with the simplification of the second term? No, Step 8 had the second term fully differentiated. Step 9 differentiates the first term and simplifies the second term's coefficient? Step 8: `Derivative(log(3*x), x) - (1/2)*(1/(1 - 9*x**2))*(-18*x)`. Step 9: `1/x - (1/2)*(1/(1 - 9*x**2))*(-18*x)`. The second term is unchanged algebraically, just the first term is differentiated. So Step 9 is purely differentiating the first term. The label 'derivative' is used. But 'derivative' is defined as unfolding d/dx on known forms (x, constant, sin, exp). Log is not in that list. The vocabulary has 'logarithmic'. Therefore, the label should be 'logarithmic' or 'chain'. Using 'derivative' is a mislabeling defect.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-28 — [domain objection, downgraded to style] Step 9 claims to differentiate the first term using the chain rule, but the result 1/x is incorrect for d/dx(log(3x)) which should be 3/(3x) = 1/x only if the constant factor is handled, but the label 'derivative' is used for a step that actually performs differentiation of a composite function log(3x) -> 1/(3x)*3 = 1/x. While the result is correct, the step skips the explicit chain rule application for the first term or mislabels the simplification. More critically, Step 9 applies differentiation to the first term while the second term was already differentiated in Step 8. The transition from Step 8 to Step 9 differentiates the first term `Derivative(log(3*x), x)` into `1/x`. This is a differentiation step. However, the note says 'using the chain rule'. The derivative of log(3x) is indeed 1/x. But look at Step 9's label: 'derivative'. This is acceptable. Let's look closer at Step 2. Step 2 splits log(A/B) into log(A) - log(B). This is a logarithmic property. The label is 'simplify'. This is acceptable. Step 4 brings the exponent 1/2 out. Label 'constant-multiple'. Acceptable. Step 7 applies chain rule to log(1-9x^2). Label 'chain'. Acceptable. Step 8 differentiates 1-9x^2. Label 'derivative'. Acceptable. Step 9 differentiates log(3x). The label is 'derivative'. The note says 'using the chain rule'. Is this a defect? The rule 'derivative' covers unfolding d/dx on known forms. log(3x) is not a 'known form' in the base list (x, constant, sin, exp). It requires chain rule or log properties. If the system considers log(kx) a known form, it's fine. But typically, one would use 'chain' or 'logarithmic' (not in list) or 'simplify' (log(3x)=log3+logx). The step jumps from Derivative(log(3x), x) to 1/x. This is a differentiation step. The label 'derivative' is in the vocabulary. Is it wrong? The prompt says 'derivative' means unfolding d/dx on a known form. Is log(3x) a known form? Usually, known forms are x, c, sin(x), exp(x). log(x) is not explicitly listed in the 'derivative' definition text provided in the prompt ('derivative of x, of a constant, of sin, of exp'). It does not list log. Therefore, applying 'derivative' to log(3x) is likely a defect because 'logarithmic' is not in the vocabulary, and 'chain' is not applied explicitly in the label (though mentioned in note). Wait, the vocabulary includes 'logarithmic'. The prompt says: 'Each step names its rule from this fixed vocabulary... logarithmic'. Step 9 uses 'derivative' but the operation is differentiating a logarithm. The rule for differentiating log is 'logarithmic' or 'chain' if composite. Since 'logarithmic' is in the vocabulary, using 'derivative' for a log function is incorrect labeling. The step should likely be 'chain' (if viewing as log(u)) or 'logarithmic' (if viewing as log rule). Given 'logarithmic' is in the list, failing to use it for a log derivative is a labeling error. Also, Step 9 combines the differentiation of the first term with the simplification of the second term? No, Step 8 had the second term fully differentiated. Step 9 differentiates the first term and simplifies the second term's coefficient? Step 8: `Derivative(log(3*x), x) - (1/2)*(1/(1 - 9*x**2))*(-18*x)`. Step 9: `1/x - (1/2)*(1/(1 - 9*x**2))*(-18*x)`. The second term is unchanged algebraically, just the first term is differentiated. So Step 9 is purely differentiating the first term. The label 'derivative' is used. But 'derivative' is defined as unfolding d/dx on known forms (x, constant, sin, exp). Log is not in that list. The vocabulary has 'logarithmic'. Therefore, the label should be 'logarithmic' or 'chain'. Using 'derivative' is a mislabeling defect.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 9 claims to use the 'derivative' rule, but it applies the chain rule to differentiate log(3*x). The 'derivative' rule is reserved for basic forms (like d/dx log(x) = 1/x), whereas differentiating a composite function like log(3*x) requires the 'chain' rule.gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.