Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(2 x + 4 \right)}}{4} \)
Problem 2.718 · hard Video
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x + 4 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(2 x + 4 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(2 x + 4 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(2 x + 4 \right)}}{4} \]constantFactor out the constant 1/4.✓ Proved
- \[ = - \frac{\cos{\left(2 x + 4 \right)} \frac{d}{d x} \left(2 x + 4\right)}{4} + \frac{d}{d x} \frac{x}{2} \]trigApply the derivative of the sine function.✓ Proved
- \[ = - \frac{\cos{\left(2 x + 4 \right)} \frac{d}{d x} 2 x}{4} + \frac{d}{d x} \frac{x}{2} \]sumApply the sum rule to the inner function.✓ Proved
- \[ = - \frac{\cos{\left(2 x + 4 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]constant-multiple algebraFactor out the constant 2. Simplify the constant coefficients.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(2 x + 4 \right)}}{2} \]derivative algebraDifferentiate the remaining terms. Simplify the expression.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(2 x + 4 \right)}}{2} \)
✓ 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 |
| 9 | ✓ 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 (error) — Step 5 replaces Derivative(2*x + 4, x) with Derivative(2*x, x), omitting the derivative of the constant 4. This is not a valid application of the sum rule and introduces an error in the differentiation process.qwen3.6:27b-mlx: fail (style) — Step 5 is labeled 'sum' but applies the constant rule (derivative of a sum is the sum of derivatives, specifically dropping the constant term 4). Step 3 is labeled 'constant' but effectively applies the constant multiple rule; while 'constant-multiple' is in the vocabulary, 'constant' is often used for factoring out constants in this context, but Step 5's label is clearly wrong for the operation performed (differentiating a constant term to zero is part of the sum rule application, but the step shows Derivative(2*x+4) becoming Derivative(2*x), which implies the derivative of 4 was taken and dropped, or it's just rewriting the argument. Actually, looking closely at Step 5: Derivative(2*x + 4, x) becomes Derivative(2*x, x). This step removes the '+4' from inside the derivative operator. This is not the sum rule applied to the outer expression. It is an application of the sum rule *inside* the derivative, or simply recognizing that the derivative of a constant is 0. However, the step shows `Derivative(2*x + 4, x)` turning into `Derivative(2*x, x)`. This is mathematically incorrect as a single step unless it implies `Derivative(2*x, x) + Derivative(4, x)` and then `Derivative(4, x)` is 0. But the step just deletes the 4. This is a 'simplify' or 'algebra' step on the argument, or a 'sum' rule application followed by a 'derivative' rule application. Applying two rules (sum rule to split, then derivative rule to kill constant) in one step is a defect. Also, the label 'sum' is misleading for a step that just drops a constant inside a derivative without showing the split.
Senior review claude-sonnet-5-5, 2026-10-12: pass — Every line was proved equivalent by SymPy, and the rule names fit the steps. Step 5 skips an explicit Derivative(4, x) = 0 line, but that is a compression of the sum-rule step and not a defect.
gpt-oss:20b: dismiss — The derivative of the constant 4 is 0, so Derivative(2*x+4, x) = Derivative(2*x, x) is a true equality (SymPy proved it), and dropping the constant term is a legitimate use of the sum rule.qwen3.6:27b-mlx: dismiss — The 'sum' label names a rule the step actually used, since the constant term contributes nothing to the sum. Dropping the zero-derivative term is part of that same change, so this is not two independent operations; the step is also mathematically correct.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-12 — Every line was proved equivalent by SymPy, and the rule names fit the steps. Step 5 skips an explicit Derivative(4, x) = 0 line, but that is a compression of the sum-rule step and not a defect.qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 5 is labeled 'sum' but applies the constant rule (derivative of a sum is the sum of derivatives, specifically dropping the constant term 4). Step 3 is labeled 'constant' but effectively applies the constant multiple rule; while 'constant-multiple' is in the vocabulary, 'constant' is often used for factoring out constants in this context, but Step 5's label is clearly wrong for the operation performed (differentiating a constant term to zero is part of the sum rule application, but the step shows Derivative(2*x+4) becoming Derivative(2*x), which implies the derivative of 4 was taken and dropped, or it's just rewriting the argument. Actually, looking closely at Step 5: Derivative(2*x + 4, x) becomes Derivative(2*x, x). This step removes the '+4' from inside the derivative operator. This is not the sum rule applied to the outer expression. It is an application of the sum rule *inside* the derivative, or simply recognizing that the derivative of a constant is 0. However, the step shows `Derivative(2*x + 4, x)` turning into `Derivative(2*x, x)`. This is mathematically incorrect as a single step unless it implies `Derivative(2*x, x) + Derivative(4, x)` and then `Derivative(4, x)` is 0. But the step just deletes the 4. This is a 'simplify' or 'algebra' step on the argument, or a 'sum' rule application followed by a 'derivative' rule application. Applying two rules (sum rule to split, then derivative rule to kill constant) in one step is a defect. Also, the label 'sum' is misleading for a step that just drops a constant inside a derivative without showing the split.gpt-oss:20b: fail (error) 2026-09-26 — Step 5 replaces Derivative(2*x + 4, x) with Derivative(2*x, x), omitting the derivative of the constant 4. This is not a valid application of the sum rule and introduces an error in the differentiation process.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies the sum rule to the inner function 2*x+4 but then drops the derivative of the constant 4 (which is 0). This combines two operations—applying the sum rule and simplifying the zero term—in a single step, violating the rule‑by‑step requirement.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the sum rule, constant multiple rule, chain rule, and basic derivative rules in a logical sequence. Each step changes only one aspect of the expression and uses an acceptable label from the provided vocabulary.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 omits the derivative of the constant 4, and step 8 incorrectly differentiates cos(2*x+4) as a constant. The solution applies more than one rule at once and contains false differentiation steps.
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.