Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(4 x - 3 \right)} \right)}}{4} \)
Problem 2.1903 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(4 x - 3 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(4 x - 3 \right)} \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x - 3 \right)} \right)}}{4} \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos{\left(4 x - 3 \right)}}{4 \cos{\left(4 x - 3 \right)}} \]chain algebraApply the chain rule to the cosine function. Combine the fraction terms.✓ Proved
- \[ = - \frac{\sin{\left(4 x - 3 \right)} \frac{d}{d x} \left(4 x - 3\right)}{4 \cos{\left(4 x - 3 \right)}} \]trigDifferentiate the cosine function.✓ Proved
- \[ = - \frac{\sin{\left(4 x - 3 \right)}}{\cos{\left(4 x - 3 \right)}} \]derivative algebra simplifyDifferentiate the inner linear function. Multiply the terms together. Cancel the common factor of 4.✓ Proved
- \[ = - \tan{\left(4 x - 3 \right)} \]trigUse the identity sin(u)/cos(u) = tan(u).✓ Proved
Answer \( - \tan{\left(4 x - 3 \right)} \)
Mind the domain. The answer is also defined at points 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(4*x - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 3) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 3) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 3) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 3) = 0 tan has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 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) — Step 2 is labeled 'logarithmic' but the step shown is merely the linearity of the derivative (pulling out the constant 1/4), which was already handled in Step 1. The actual application of the logarithmic derivative rule occurs in Step 3. This mislabeling conflates linearity with the specific derivative rule for log.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (style) 2026-10-09 — Step 2 is labeled 'logarithmic' but the step shown is merely the linearity of the derivative (pulling out the constant 1/4), which was already handled in Step 1. The actual application of the logarithmic derivative rule occurs in Step 3. This mislabeling conflates linearity with the specific derivative rule for log.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 2 is labeled 'logarithmic' but does not apply the logarithmic derivative rule; it merely rewrites the expression. The actual application of the logarithmic derivative rule occurs in Step 3, which is incorrectly labeled 'chain'. Step 2 should be 'rewrite' or 'algebra', and Step 3 should be 'logarithmic' (or 'chain' if viewed as part of the chain, but 'logarithmic' is the specific rule for d/dx log(u)). More critically, Step 2 claims to apply the rule but changes nothing structurally other than moving the constant, which was already done in Step 1. The label 'logarithmic' implies d/dx log(u) = u'/u, which is not what happens in Step 2. Step 3 performs the differentiation of the log, so Step 3 should be labeled 'logarithmic' (or 'chain' if focusing on the outer function, but 'logarithmic' is more precise for the log part). However, the contract says 'logarithmic' is a valid label. The main issue is that Step 2 does not apply the logarithmic rule; it just sits there. Step 3 applies the chain rule to the log function (d/dx log(u) = 1/u * u'). So Step 3 is the one applying the log derivative. Labeling Step 2 as 'logarithmic' is wrong because no differentiation happens there. Labeling Step 3 as 'chain' is acceptable if it refers to the chain rule applied to the composite function log(cos(...)), but typically 'logarithmic' covers the d/dx log(u) part. Let's look closer. Step 2: 1/4 * Derivative(log(cos(4*x - 3)), x). This is just a presentation step. It should be 'rewrite' or 'algebra'. Calling it 'logarithmic' is a defect because the rule isn't applied. Step 3: 1/4 * (1/(cos(4*x - 3))) * Derivative(cos(4*x - 3), x). This applies d/dx log(u) = 1/u * u'. This is the logarithmic rule combined with the chain rule. The label 'chain' is acceptable if it refers to the chain rule aspect, but 'logarithmic' is also acceptable. However, Step 2 is definitely mislabeled. It does not apply the logarithmic rule. It just moves the constant. Wait, Step 1 pulled out the constant. Step 2 is just... there? It's identical to the RHS of Step 1 except for the label. Step 1: Derivative(log(...)/4, x). Step 2: 1/4 * Derivative(log(...), x). This is the constant-multiple rule. Step 1 is labeled 'constant-multiple'. Step 2 is labeled 'logarithmic'. Step 2 does not apply the logarithmic rule. It is a no-op or a rewrite. This is a defect.gpt-oss:20b: fail (style) 2026-10-09 — Step 5 incorrectly labels the application of the chain rule as "trig"; it should be "chain". No other steps combine multiple rules or contain mathematical errors.
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.