Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(10 x + 2 \right)}}{20} \)
Problem 2.878 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x + 2 \right)}}{20} \).
- \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(10 x + 2 \right)}}{20}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(10 x + 2 \right)}}{20} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(10 x + 2 \right)}}{20} \]constantFactor out the constant 1/20.✓ Proved
- \[ = \frac{\cos{\left(10 x + 2 \right)} \frac{d}{d x} \left(10 x + 2\right)}{20} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]trigApply the derivative of the sine function.✓ Proved
- \[ = \frac{\cos{\left(10 x + 2 \right)} \frac{d}{d x} 10 x}{20} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]constantApply the sum rule to the inner function.✓ Proved
- \[ = \frac{\cos{\left(10 x + 2 \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]derivative algebraDifferentiate the linear term 10*x. Simplify the fraction 10/20.✓ Proved
- \[ = \frac{\cos{\left(10 x + 2 \right)}}{2} - \frac{1}{2} \]derivativeDifferentiate the remaining term -x/2.✓ Proved
Answer \( \frac{\cos{\left(10 x + 2 \right)} - 1}{2} \)
✓ 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. How we verify
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 |
| 8 | ✓ 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 removal of the constant term in the inner derivative. The correct label should be "derivative" (derivative of the constant 2 is 0), not "constant" or "sum". This mislabeling could mislead a student about which rule applies to eliminating constants inside a derivative.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] Step 5 is labeled 'constant' but the note describes applying the 'sum rule' to the inner function, which is a contradiction. Additionally, Step 5 applies the sum rule to the inner derivative, but the transition from Step 4 to 5 only shows the result of that sum rule on the '10*x' term while implicitly handling the constant '2' (derivative 0) without showing the split, or rather, it skips the explicit sum rule application for the inner term and jumps to the derivative of 10x. More critically, Step 5's label 'constant' does not match the operation described in the note ('sum rule') nor the algebraic change (which is actually applying the sum rule inside the derivative operator, or rather, the step is ill-defined because Derivative(10*x+2, x) becomes Derivative(10*x, x) + Derivative(2, x), but the step only shows one term. Actually, looking closely: Step 4 has `Derivative(10*x + 2, x)`. Step 5 has `Derivative(10*x, x)`. The term `Derivative(2, x)` which is 0 is dropped. This is a valid simplification, but the label 'constant' is wrong for a sum rule application, and the note says 'sum rule'. The label 'constant' usually refers to pulling out constants or derivative of a constant. Here, the step is effectively `Derivative(A+B) -> Derivative(A) + Derivative(B)` then `Derivative(B)=0`. This is two steps compressed or a mislabeled step. The note says 'Apply the sum rule', so the label should be 'sum'. The label 'constant' is incorrect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-27 — [domain objection, downgraded to style] Step 5 is labeled 'constant' but the note describes applying the 'sum rule' to the inner function, which is a contradiction. Additionally, Step 5 applies the sum rule to the inner derivative, but the transition from Step 4 to 5 only shows the result of that sum rule on the '10*x' term while implicitly handling the constant '2' (derivative 0) without showing the split, or rather, it skips the explicit sum rule application for the inner term and jumps to the derivative of 10x. More critically, Step 5's label 'constant' does not match the operation described in the note ('sum rule') nor the algebraic change (which is actually applying the sum rule inside the derivative operator, or rather, the step is ill-defined because Derivative(10*x+2, x) becomes Derivative(10*x, x) + Derivative(2, x), but the step only shows one term. Actually, looking closely: Step 4 has `Derivative(10*x + 2, x)`. Step 5 has `Derivative(10*x, x)`. The term `Derivative(2, x)` which is 0 is dropped. This is a valid simplification, but the label 'constant' is wrong for a sum rule application, and the note says 'sum rule'. The label 'constant' usually refers to pulling out constants or derivative of a constant. Here, the step is effectively `Derivative(A+B) -> Derivative(A) + Derivative(B)` then `Derivative(B)=0`. This is two steps compressed or a mislabeled step. The note says 'Apply the sum rule', so the label should be 'sum'. The label 'constant' is incorrect.gpt-oss:20b: fail (style) 2026-09-27 — Step 5 incorrectly labels the removal of the constant term in the inner derivative. The correct label should be "derivative" (derivative of the constant 2 is 0), not "constant" or "sum". This mislabeling could mislead a student about which rule applies to eliminating constants inside a derivative.qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 5 is labeled 'constant' but the note describes applying the sum rule to the inner function; the label does not match the operation described. Step 4 is labeled 'trig' but applies the chain rule (differentiating the inner function), so 'chain' would be the more precise label from the vocabulary, though 'trig' is arguably acceptable if interpreted broadly as the derivative of sine. However, the mismatch in Step 5 is a clear defect.gpt-oss:20b: fail (style) 2026-09-26 — Step 5 applies the sum rule to the inner function 10*x+2 but labels it "constant". The correct label should be "sum" (or "constant-multiple" for the 10*x part).qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 5 is labeled 'constant' but applies the sum rule to the inner function (10*x + 2). Step 3 is labeled 'constant' but effectively applies the constant-multiple rule; while 'constant' is in the vocabulary, 'constant-multiple' is more precise, but the primary defect is the mislabeling of Step 5.gpt-oss:20b: pass 2026-09-26
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 2026-09-27 with SymPy 1.14.0.