Derivative of \( \displaystyle x - \frac{\sin{\left(10 x - 6 \right)}}{10} \)
Problem 2.1430 · hard
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(10 x - 6 \right)}}{10} \).
- \[ \frac{d}{d x} \left(x - \frac{\sin{\left(10 x - 6 \right)}}{10}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x - \frac{d}{d x} \frac{\sin{\left(10 x - 6 \right)}}{10} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} x - \frac{\frac{d}{d x} \sin{\left(10 x - 6 \right)}}{10} \]constantFactor out the constant 1/10.✓ Proved
- \[ = 1 - \frac{\frac{d}{d x} \sin{\left(10 x - 6 \right)}}{10} \]derivativeThe derivative of x is 1.✓ Proved
- \[ = - \frac{\cos{\left(10 x - 6 \right)} \frac{d}{d x} \left(10 x - 6\right)}{10} + 1 \]trigApply the chain rule to the sine function.✓ Proved
- \[ = 1 - \cos{\left(10 x - 6 \right)} \]derivative simplifyThe derivative of 10*x - 6 is 10. Simplify the expression by canceling the 10.✓ Proved
Answer \( 1 - \cos{\left(10 x - 6 \right)} \)
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 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 5 incorrectly labels the application of the chain rule as "trig". The correct label should be "chain" (or simply "derivative" if the vocabulary allows), since the step applies the chain rule to the sine function, not a pure trigonometric simplification. This mislabeling could mislead a student about the rule being used.qwen3.6:27b-mlx: fail (misleading) — Step 5 applies the chain rule but is labeled 'trig'. The label 'trig' typically refers to trigonometric identities or basic derivatives, whereas the step explicitly invokes the chain rule structure (derivative of outer function times derivative of inner function). While 'chain' is in the vocabulary, using 'trig' obscures the specific rule applied. More critically, Step 3 is labeled 'constant' but performs factoring of a constant multiple, which should be labeled 'constant-multiple' to distinguish it from the derivative of a constant. However, the most significant defect is Step 5's label: it describes a chain rule application but uses the label 'trig', which is imprecise and potentially misleading given 'chain' is an available and more accurate label in the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (misleading) 2026-10-03 — Step 5 applies the chain rule but is labeled 'trig'. The label 'trig' typically refers to trigonometric identities or basic derivatives, whereas the step explicitly invokes the chain rule structure (derivative of outer function times derivative of inner function). While 'chain' is in the vocabulary, using 'trig' obscures the specific rule applied. More critically, Step 3 is labeled 'constant' but performs factoring of a constant multiple, which should be labeled 'constant-multiple' to distinguish it from the derivative of a constant. However, the most significant defect is Step 5's label: it describes a chain rule application but uses the label 'trig', which is imprecise and potentially misleading given 'chain' is an available and more accurate label in the fixed vocabulary.gpt-oss:20b: fail (style) 2026-10-03 — Step 5 incorrectly labels the application of the chain rule as "trig". The correct label should be "chain" (or simply "derivative" if the vocabulary allows), since the step applies the chain rule to the sine function, not a pure trigonometric simplification. This mislabeling could mislead a student about the rule being used.qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 5 is labeled 'trig' but the note and the operation describe applying the chain rule; the label should be 'chain'. Step 3 is labeled 'constant' but factors out a constant from a derivative, which is better described as 'constant-multiple'.gpt-oss:20b: fail (style) 2026-10-03 — Step 5 incorrectly labels the application of the chain rule as "trig". The correct label from the allowed vocabulary is "chain". No other steps violate the contract.
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 2026-10-03 with SymPy 1.14.0.