Derivative of \( \displaystyle \frac{\ln{\left(\sin{\left(2 x - 3 \right)} \right)}}{2} \)
Problem 2.1497 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(2 x - 3 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\sin{\left(2 x - 3 \right)} \right)}}{2} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x - 3 \right)} \right)}}{2} \]chainApply the chain rule to the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(2 x - 3 \right)}}{2 \sin{\left(2 x - 3 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{\cos{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \sin{\left(2 x - 3 \right)}} \]derivativeDifferentiate the inner linear function.✓ Proved
- \[ = \frac{\cos{\left(2 x - 3 \right)}}{\sin{\left(2 x - 3 \right)}} \]algebra simplifySimplify the expression by multiplying constants. Combine the terms into a single fraction.✓ Proved
- \[ = \cot{\left(2 x - 3 \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( \frac{1}{\tan{\left(2 x - 3 \right)}} \)
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 sin(2*x - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 3) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 3) = 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(2*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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — Step 2 is labeled 'chain', but the step shown is the application of the chain rule to the logarithm, which is correctly handled in Step 3. Step 2 actually just pulls out the constant 1/2, which was already done in Step 1, or it is a redundant step that does nothing. More critically, Step 2 claims to apply the chain rule to the natural logarithm, but the expression on the RHS is still `Derivative(log(...), x)`. The chain rule application happens in Step 3. Step 2 is effectively a no-op or a mislabeled step that does not change the expression. Actually, looking closely: Step 1 is `Derivative(log(...)/2, x)`. Step 2 is `1/2 * Derivative(log(...), x)`. This is the constant multiple rule. The label is 'chain'. This is a labeling error. Step 2 applies the constant multiple rule, not the chain rule. The chain rule is applied in Step 3. Therefore, Step 2 has the wrong label.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.