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
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 \( 3 e^{2 x - 1} \sin{\left(2 x - 1 \right)} \)
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 |
| 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: fail (error) — 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) — 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.
Every verdict on record (4)
qwen3.6:27b-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 2026-09-26 with SymPy 1.14.0.