Derivative of \( \displaystyle \frac{\left(3 \sin{\left(2 x - 1 \right)} - 3 \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \)
Problem 2.874 · hard Video
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \).
- \[ \frac{d}{d x} \frac{\left(3 \sin{\left(2 x - 1 \right)} - 3 \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \]constant-multiplePull out the constant coefficient.✓ Proved
- \[ = \frac{3 \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) \frac{d}{d x} e^{2 x - 1}}{4} + \frac{3 e^{2 x - 1} \frac{d}{d x} \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right)}{4} \]productApply the product rule.✓ Proved
- \[ = \left(\frac{3 \sin{\left(2 x - 1 \right)}}{4} - \frac{3 \cos{\left(2 x - 1 \right)}}{4}\right) \frac{d}{d x} e^{2 x - 1} + \left(\frac{3 \frac{d}{d x} \sin{\left(2 x - 1 \right)}}{4} - \frac{3 \frac{d}{d x} \cos{\left(2 x - 1 \right)}}{4}\right) e^{2 x - 1} \]sumApply the difference rule to the first term.✓ Proved
- \[ = \left(\frac{3 \sin{\left(2 x - 1 \right)}}{4} - \frac{3 \cos{\left(2 x - 1 \right)}}{4}\right) \frac{d}{d x} e^{2 x - 1} + \left(\frac{3 \sin{\left(2 x - 1 \right)}}{2} + \frac{3 \cos{\left(2 x - 1 \right)}}{2}\right) e^{2 x - 1} \]trig algebraDifferentiate the sine and cosine terms using the chain rule. Simplify the signs inside the parenthesis.✓ Proved
- \[ = 2 \left(\frac{3 \sin{\left(2 x - 1 \right)}}{4} - \frac{3 \cos{\left(2 x - 1 \right)}}{4}\right) e^{2 x - 1} + \left(\frac{3 \sin{\left(2 x - 1 \right)}}{2} + \frac{3 \cos{\left(2 x - 1 \right)}}{2}\right) e^{2 x - 1} \]exponentialDifferentiate the exponential term.✓ Proved
- \[ = 3 e^{2 x - 1} \sin{\left(2 x - 1 \right)} \]algebra algebra simplify simplifyFactor out the common exponential term. Distribute the 2 in the second part of the expression. Combine like terms inside the parenthesis. Multiply the remaining constants and simplify.✓ Proved
Answer \( 3 e^{2 x - 1} \sin{\left(2 x - 1 \right)} \)
✓ 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ 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: passqwen3.6:27b-mlx: fail (error) — Step 4 applies the distributive property (algebra) to split the product term, but is labeled 'sum'. While 'sum' is in the vocabulary, it typically refers to the linearity of the derivative operator (d/dx(f+g) = d/dx(f)+d/dx(g)), not the algebraic distribution of a constant or function over a sum. More critically, Step 5 is labeled 'trig' but performs differentiation of composite functions (chain rule) for sin and cos. The label 'trig' is not in the allowed vocabulary for differentiation rules (only 'chain' is), and 'trig' usually implies trigonometric identities, not differentiation. The step applies the chain rule, so it should be labeled 'chain'. Furthermore, Step 5 differentiates two terms at once, violating the 'one thing per step' rule if interpreted strictly, but the primary defect is the incorrect label 'trig' for a differentiation step where 'chain' is the required label from the fixed vocabulary.
Senior review claude-sonnet-5-5, 2026-10-13: pass — Every line was proved equivalent by SymPy and the rule labels are all legitimate. The final result 3*exp(2x-1)*sin(2x-1) is correct, since 3/4 * exp * (4 sin) simplifies to it.
qwen3.6:27b-mlx: dismiss — Step 4 splits the derivative of a difference into a difference of derivatives, which is the sum rule, so 'sum' is correct. 'trig' is in the allowed vocabulary and fits differentiating sin and cos, with 'chain' also acceptable. Handling sin and cos with the same derivative rule in one step is not two independent changes.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — Every line was proved equivalent by SymPy and the rule labels are all legitimate. The final result 3*exp(2x-1)*sin(2x-1) is correct, since 3/4 * exp * (4 sin) simplifies to it.qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 4 applies the distributive property (algebra) to split the product term, but is labeled 'sum'. While 'sum' is in the vocabulary, it typically refers to the linearity of the derivative operator (d/dx(f+g) = d/dx(f)+d/dx(g)), not the algebraic distribution of a constant or function over a sum. More critically, Step 5 is labeled 'trig' but performs differentiation of composite functions (chain rule) for sin and cos. The label 'trig' is not in the allowed vocabulary for differentiation rules (only 'chain' is), and 'trig' usually implies trigonometric identities, not differentiation. The step applies the chain rule, so it should be labeled 'chain'. Furthermore, Step 5 differentiates two terms at once, violating the 'one thing per step' rule if interpreted strictly, but the primary defect is the incorrect label 'trig' for a differentiation step where 'chain' is the required label from the fixed vocabulary.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 applies the distributive property of multiplication over addition (algebra) to split the product rule result, but labels it 'sum'. The 'sum' rule applies to the derivative operator (d/dx(u+v) = d/dx(u) + d/dx(v)), not to distributing a constant or function factor across a sum. Additionally, Step 5 labels the differentiation of sin and cosine as 'trig', but the step involves the chain rule (differentiating the inner function 2x-1), so 'chain' is the required label for the differentiation rule applied.gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies two derivative rules at once (for sin and cos) but is labeled as a single "trig" step, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 5 applies the chain rule to differentiate the sine and cosine terms but labels the step 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for differentiation. Since 'chain' is in the vocabulary, the step should be labeled 'chain' (or 'derivative' if 'chain' is considered a sub-case, but 'trig' is definitely incorrect for a differentiation step).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.