Derivative of \( \displaystyle \ln{\left(\sin{\left(x + 1 \right)} \right)} \)
Problem 2.1583 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\sin{\left(x + 1 \right)} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\sin{\left(x + 1 \right)} \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(x + 1 \right)}}{\sin{\left(x + 1 \right)}} \]logarithmicApply the rule for the derivative of a logarithm.✓ Proved
- \[ = \frac{\cos{\left(x + 1 \right)}}{\sin{\left(x + 1 \right)}} \]derivativeDifferentiate the inner function sin(x + 1).✓ Proved
- \[ = \frac{1}{\tan{\left(x + 1 \right)}} \]simplifySimplify the ratio of cosine to sine.✓ Proved
Answer \( \frac{1}{\tan{\left(x + 1 \right)}} \)
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 undefined where sin(x + 1) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x + 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x + 1) = 0 tan has poles at odd multiples of pi/2 undefined where tan(x + 1) = 0 |
| 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(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: passqwen3.6:27b-mlx: fail (error) — Step 3 applies the chain rule to differentiate sin(x + 1), but is labeled 'derivative'. The label 'derivative' is reserved for basic derivatives (e.g., d/dx sin(u) -> cos(u) * u' is not a basic derivative; d/dx sin(x) -> cos(x) is). Differentiating a composite function requires the 'chain' rule. Additionally, Step 2 is labeled 'logarithmic' which is acceptable, but Step 3's failure to use 'chain' is a defect in rule naming for a composite function differentiation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 applies the chain rule to differentiate sin(x + 1), but is labeled 'derivative'. The label 'derivative' is reserved for basic derivatives (e.g., d/dx sin(u) -> cos(u) * u' is not a basic derivative; d/dx sin(x) -> cos(x) is). Differentiating a composite function requires the 'chain' rule. Additionally, Step 2 is labeled 'logarithmic' which is acceptable, but Step 3's failure to use 'chain' is a defect in rule naming for a composite function differentiation.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 applies the chain rule to differentiate sin(x + 1) but is labeled 'derivative'. The label 'derivative' is reserved for unfolding d/dx on a known form (like d/dx sin(u) = cos(u) * u'), but here the inner function x+1 also requires differentiation, necessitating the 'chain' rule label. Additionally, the transition from step 2 to 3 implicitly differentiates the inner function without explicitly showing the chain rule application or labeling it as such, violating the one-rule-per-step constraint if interpreted strictly, but primarily the label is incorrect for the operation performed.gpt-oss:20b: fail (style) 2026-10-05 — Step 1 uses an unlabelled rule. Every step must name a rule from the allowed vocabulary; the derivative of the logarithm should be applied immediately (e.g., “logarithmic”) rather than leaving the step unlabeled.
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 2026-10-05 with SymPy 1.14.0.