Derivative of \( \displaystyle \frac{3 \ln{\left(\sin{\left(4 x - 1 \right)} \right)}}{4} \)
Problem 2.1980 · hard
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\sin{\left(4 x - 1 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \frac{3 \ln{\left(\sin{\left(4 x - 1 \right)} \right)}}{4} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\sin{\left(4 x - 1 \right)} \right)}}{4} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \sin{\left(4 x - 1 \right)}}{4 \sin{\left(4 x - 1 \right)}} \]derivativeDifferentiate the natural logarithm function.✓ Proved
- \[ = \frac{3 \cos{\left(4 x - 1 \right)} \frac{d}{d x} \left(4 x - 1\right)}{4 \sin{\left(4 x - 1 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{3 \cos{\left(4 x - 1 \right)}}{\sin{\left(4 x - 1 \right)}} \]derivative algebraDifferentiate the linear expression inside the cosine. Multiply the constants.✓ Proved
- \[ = 3 \cot{\left(4 x - 1 \right)} \]simplifySimplify the ratio of cosine to sine.✓ Proved
Answer \( \frac{3}{\tan{\left(4 x - 1 \right)}} \)
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 sin(4*x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x - 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x - 1) = 0 cot has poles at multiples of pi |
| 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 undefined where tan(4*x - 1) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — Step 2 incorrectly labels the operation as "chain"; it merely factors out the constant 3/4, which is an algebraic simplification, not a chain rule application. The correct label for that step should be "algebra".qwen3.6:27b-mlx: fail (style) — Step 2 is labeled 'chain' but performs the constant-multiple rule (pulling out 3/4); the chain rule is applied in Step 3. Step 5 is labeled 'derivative' but the note incorrectly refers to differentiating an expression inside a cosine, whereas it is inside a sine.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-10 — Step 2 incorrectly labels the operation as "chain"; it merely factors out the constant 3/4, which is an algebraic simplification, not a chain rule application. The correct label for that step should be "algebra".qwen3.6:27b-mlx: fail (style) 2026-10-10 — Step 2 is labeled 'chain' but performs the constant-multiple rule (pulling out 3/4); the chain rule is applied in Step 3. Step 5 is labeled 'derivative' but the note incorrectly refers to differentiating an expression inside a cosine, whereas it is inside a sine.qwen3.6:27b-mlx: fail (misleading) 2026-10-10 — Step 2 is labeled 'chain', but the transformation from line 1 to line 2 only factors out the constant 3/4, which is a 'constant-multiple' step. The chain rule is not applied until step 3. This mislabels the operation performed.gpt-oss:20b: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.