Derivative of \( \displaystyle - \ln{\left(\sin{\left(x - 3 \right)} \right)} \)
Problem 2.1896 · hard
Differentiate \( \displaystyle f(x) = - \ln{\left(\sin{\left(x - 3 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\sin{\left(x - 3 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\sin{\left(x - 3 \right)} \right)} \]constantPull out the constant factor -1.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sin{\left(x - 3 \right)}}{\sin{\left(x - 3 \right)}} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\cos{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right)}{\sin{\left(x - 3 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = - \frac{\cos{\left(x - 3 \right)}}{\sin{\left(x - 3 \right)}} \]derivative algebraDifferentiate the inner function x - 3. Simplify the expression.✓ Proved
- \[ = - \frac{1}{\tan{\left(x - 3 \right)}} \]simplifyRewrite using the cotangent identity.✓ Proved
Answer \( - \frac{1}{\tan{\left(x - 3 \right)}} \)
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.
✓ 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(x - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0 tan has poles at odd multiples of pi/2 undefined where tan(x - 3) = 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 - 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)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09qwen3.6:27b-mlx: fail (style) 2026-10-09 — Step 2 is labeled 'constant' but applies the constant-multiple rule (pulling out -1); 'constant' typically refers to the derivative of a constant being zero. Step 3 is labeled 'logarithmic' but the note claims it applies the chain rule; while the step combines the log derivative and chain rule, the label 'logarithmic' is acceptable for the log part, but the note is misleadingly specific about 'chain rule' when the step also handles the log derivative structure. More critically, Step 3 applies the derivative of log(u) as u'/u, which is a combination of the logarithmic rule and the chain rule. The contract requires one rule per step. Step 3 applies the logarithmic differentiation formula (which inherently includes the chain rule for the argument) in one go. However, looking closely at the vocabulary, 'logarithmic' is a valid label for d/dx log(u) = u'/u. The issue is that Step 3 computes the derivative of log(sin(...)) entirely, resulting in (cos(...)/sin(...)) * 1? No, Step 3 results in Derivative(sin(x-3), x) / sin(x-3). This is just the outer derivative of log. It has not differentiated the inner sin yet. So Step 3 is: d/dx log(u) = u'/u. This is the 'logarithmic' rule. Step 4 is: d/dx sin(v) = cos(v)*v'. This is the 'chain' rule. This seems compliant. Let's re-read Step 2. 'constant' rule usually means d/dx(c) = 0 or d/dx(cf) = c d/dx f? The vocabulary has 'constant-multiple'. Step 2 uses label 'constant'. The vocabulary list includes 'constant-multiple'. It does NOT include 'constant' as a rule for pulling out factors? Wait, the list is: algebra, chain, constant, constant-multiple... 'constant' is in the list. Usually 'constant' means derivative of a constant. 'constant-multiple' means pulling out a constant. Step 2 pulls out -1. It should be labeled 'constant-multiple'. Labeling it 'constant' is a defect because 'constant' is a distinct label in the vocabulary (likely for d/dx(c)=0). Using 'constant' for linearity is a mislabeling.gpt-oss:20b: pass 2026-10-09
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.