Derivative of \( \displaystyle \frac{5 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \)
Problem 2.3 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sin{\left(2 x \right)}}{2 \sin{\left(2 x \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \cos{\left(2 x \right)} \frac{d}{d x} 2 x}{2 \sin{\left(2 x \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{5 \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} \]derivative algebra algebraDifferentiate the innermost function 2*x. Simplify the constant terms. Combine the terms into a single fraction.✓ Proved
- \[ = 5 \cot{\left(2 x \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( \frac{5}{\tan{\left(2 x \right)}} \)
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.
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x) = 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) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
deepseek-r1:70b: passgpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and basic derivatives in a step-by-step manner. The final simplification to cotangent is valid, and although the stated answer uses tan, the steps provided are mathematically sound and adhere to the contract.
Every verdict on record (11)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule, and basic derivatives in a step-by-step manner. The final simplification to cotangent is valid, and although the stated answer uses tan, the steps provided are mathematically sound and adhere to the contract.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 1 is a no-op that merely restates the problem but is labeled 'constant-multiple', which implies a transformation occurred. The step should be labeled 'derivative' or removed entirely.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — The final answer is given as 5/tan(2*x), but the solution steps conclude with 5*cot(2*x). The step from 5*cot(2*x) to 5/tan(2*x) is missing, or the stated answer does not match the last derived line. Additionally, Step 1 is labeled 'constant-multiple' but performs no operation; it is just the initial setup.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19deepseek-r1:70b: pass 2026-09-16
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.