Derivative of \( \displaystyle \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \)
Problem 2.658 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \).
- \[ \frac{d}{d x} \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \]Start with the derivative of the function.✓ Proved
- \[ = \left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) \frac{d}{d x} \frac{e^{2 x - 1}}{4} + \frac{e^{2 x - 1} \frac{d}{d x} \left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right)}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1} \frac{d}{d x} \left(2 x - 1\right)}{4} + \frac{e^{2 x - 1} \frac{d}{d x} \left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right)}{4} \]constant-multipleFactor out the constant 1/4 from the second term.✓ Proved
- \[ = \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} + \frac{e^{2 x - 1} \frac{d}{d x} \left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right)}{4} \]derivativeDifferentiate the inner function 2*x - 1.✓ Proved
- \[ = \frac{\left(- 2 \sin{\left(2 x - 1 \right)} - 2 \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} + \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} \]trig algebraDifferentiate the sine and cosine terms using the chain rule. Factor out -2 from the first term.✓ Proved
- \[ = \frac{\left(- \sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} + \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} \]algebra algebraSimplify the first term by dividing by 2. Simplify the second term by multiplying by 2.✓ Proved
- \[ = - e^{2 x - 1} \sin{\left(2 x - 1 \right)} \]algebra simplify simplifyFactor out the common term exp(2*x - 1)/2. Combine like terms inside the parentheses. Final simplification.✓ 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.
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 5 applies the chain rule to differentiate the trigonometric terms but labels it 'trig', which is not the correct label for applying the chain rule. Additionally, Step 3 is labeled 'constant-multiple' but actually applies the chain rule to the exponential term, which is a mislabeling of the rule applied.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 5 applies the chain rule to differentiate the trigonometric terms but labels it 'trig', which is not the correct label for applying the chain rule. Additionally, Step 3 is labeled 'constant-multiple' but actually applies the chain rule to the exponential term, which is a mislabeling of the rule applied.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'constant-multiple' but applies the chain rule to the second term (introducing Derivative(2*x - 1, x)). The label does not match the operation performed.gpt-oss:20b: pass 2026-09-21
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.