Derivative of \( \displaystyle \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \)
Problem 2.1096 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) \frac{d}{d x} e^{2 x - 1}}{4} + \frac{e^{2 x - 1} \frac{d}{d x} \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right)}{4} \]chainApply the chain rule to both parts.✓ Proved
- \[ = \frac{\left(2 \sin{\left(2 x - 1 \right)} - 2 \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} + \frac{\left(2 \sin{\left(2 x - 1 \right)} + 2 \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \]trig algebraDifferentiate the sine and cosine terms. Distribute the negative sign.✓ Proved
- \[ = e^{2 x - 1} \sin{\left(2 x - 1 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Simplify the final expression.✓ Proved
Answer \( e^{2 x - 1} \sin{\left(2 x - 1 \right)} \)
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. 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 3 incorrectly labels the product rule as "chain". The derivative of the product should be identified with the product rule, not the chain rule.qwen3.6:27b-mlx: fail (error) — Step 3 applies the chain rule to compute the derivatives, but the product rule was already applied in Step 2. Step 3 should be labeled 'derivative' (unfolding d/dx on sin, cos, exp) or 'chain' if strictly differentiating the inner functions, but it combines the differentiation of the product terms which was already handled. More critically, Step 3 claims to apply the chain rule to 'both parts' of the product expansion, but the product rule structure was established in Step 2. Step 3 is actually performing the differentiation of the individual factors (sin, cos, exp) using the chain rule. However, the label 'chain' is acceptable for differentiating composite functions. The real issue is Step 3's note says 'Apply the chain rule to both parts', implying it's doing the differentiation. But Step 2 applied the product rule. Step 3 is just evaluating the derivatives inside the product rule formula. This is fine. Wait, let's look closer. Step 3: `1/4 * (Derivative(...) * ... + ... * Derivative(...))`. This is the result of the product rule. Step 3's output is `1/4 * ((2*cos...) * exp... + ... * 2*exp...)`. This step computes the actual derivatives. The label is 'chain'. This is correct because d/dx sin(2x-1) requires chain rule. Is there a multi-rule violation? Step 3 computes d/dx(sin(2x-1)), d/dx(cos(2x-1)), and d/dx(exp(2x-1)). It applies chain rule to all three. This is one rule applied to multiple terms, which is allowed under 'sum' or just multiple applications of the same rule. But wait, Step 3 also handles the sum inside the first derivative? No, Step 3's input has `Derivative(sin(2*x - 1) - cos(2*x - 1), x)`. Step 3's output has `2*cos(2*x - 1) - (-2*sin(2*x - 1))`. This step differentiated the sum `sin - cos`. Differentiating a sum requires the 'sum' rule (or linearity). Step 3 applies 'chain' to the inner functions but implicitly applies 'sum' to the outer structure `sin - cos`. Applying 'sum' and 'chain' in one step is a defect. The step should be split: first apply sum rule to separate sin and cos derivatives, then apply chain rule. Or label it 'sum' and 'chain'? The contract says 'Each step changes ONE thing. A step that applies two rules at once is a defect.' Step 3 applies the sum rule (to split sin and cos) AND the chain rule (to differentiate the arguments). Therefore, it is a defect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 applies the chain rule to compute the derivatives, but the product rule was already applied in Step 2. Step 3 should be labeled 'derivative' (unfolding d/dx on sin, cos, exp) or 'chain' if strictly differentiating the inner functions, but it combines the differentiation of the product terms which was already handled. More critically, Step 3 claims to apply the chain rule to 'both parts' of the product expansion, but the product rule structure was established in Step 2. Step 3 is actually performing the differentiation of the individual factors (sin, cos, exp) using the chain rule. However, the label 'chain' is acceptable for differentiating composite functions. The real issue is Step 3's note says 'Apply the chain rule to both parts', implying it's doing the differentiation. But Step 2 applied the product rule. Step 3 is just evaluating the derivatives inside the product rule formula. This is fine. Wait, let's look closer. Step 3: `1/4 * (Derivative(...) * ... + ... * Derivative(...))`. This is the result of the product rule. Step 3's output is `1/4 * ((2*cos...) * exp... + ... * 2*exp...)`. This step computes the actual derivatives. The label is 'chain'. This is correct because d/dx sin(2x-1) requires chain rule. Is there a multi-rule violation? Step 3 computes d/dx(sin(2x-1)), d/dx(cos(2x-1)), and d/dx(exp(2x-1)). It applies chain rule to all three. This is one rule applied to multiple terms, which is allowed under 'sum' or just multiple applications of the same rule. But wait, Step 3 also handles the sum inside the first derivative? No, Step 3's input has `Derivative(sin(2*x - 1) - cos(2*x - 1), x)`. Step 3's output has `2*cos(2*x - 1) - (-2*sin(2*x - 1))`. This step differentiated the sum `sin - cos`. Differentiating a sum requires the 'sum' rule (or linearity). Step 3 applies 'chain' to the inner functions but implicitly applies 'sum' to the outer structure `sin - cos`. Applying 'sum' and 'chain' in one step is a defect. The step should be split: first apply sum rule to separate sin and cos derivatives, then apply chain rule. Or label it 'sum' and 'chain'? The contract says 'Each step changes ONE thing. A step that applies two rules at once is a defect.' Step 3 applies the sum rule (to split sin and cos) AND the chain rule (to differentiate the arguments). Therefore, it is a defect.gpt-oss:20b: fail (style) 2026-09-29 — Step 3 incorrectly labels the product rule as "chain". The derivative of the product should be identified with the product rule, not the chain rule.qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 3 applies the chain rule to both factors of the product, violating the 'one rule per step' constraint; the chain rule applications should be separated into distinct steps or labeled more granularly.gpt-oss:20b: fail (style) 2026-09-28 — Step 3 is labeled "chain" but it applies the product rule (and inside it the chain rule). The correct label for that step should be "product" (or "sum" if splitting the derivative).qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 3 applies the chain rule to both terms of the product simultaneously, violating the 'one change per step' constraint; it should be split into separate steps for each derivative.gpt-oss:20b: pass 2026-09-28
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 2026-09-29 with SymPy 1.14.0.