Derivative of \( \displaystyle \frac{\left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \)
Problem 2.1328 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) \frac{d}{d x} e^{2 x + 1}}{4} + \frac{e^{2 x + 1} \frac{d}{d x} \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right)}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) \frac{d}{d x} e^{2 x + 1}}{4} + \frac{\left(\frac{d}{d x} \sin{\left(2 x + 1 \right)} - \frac{d}{d x} \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]sumApply the sum rule to the first term.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{2} + \frac{\left(2 \sin{\left(2 x + 1 \right)} + 2 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]chain algebraApply the chain rule to the sine and cosine terms. Simplify the signs within the first parenthesis.✓ 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.
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: passqwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to the sine and cosine terms but fails to differentiate the exponential term, which also requires the chain rule. This violates the 'one rule per step' constraint by performing partial differentiation, and the label 'chain' is incomplete because it ignores the exponential part of the expression.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 applies the chain rule to the sine and cosine terms but fails to differentiate the exponential term, which also requires the chain rule. This violates the 'one rule per step' constraint by performing partial differentiation, and the label 'chain' is incomplete because it ignores the exponential part of the expression.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 5 applies the chain rule to the sine and cosine terms but fails to apply the chain rule to the exponential term, leaving its derivative as just 2*exp(...) instead of correctly computing it as part of the chain rule application in that step. While the algebraic result is correct, the label 'chain' implies the rule was applied to all composite functions, but it was only applied to the trig terms, while the exp term's derivative was computed via 'derivative' and 'chain' implicitly or incorrectly labeled. More strictly, Step 5 computes Derivative(exp(2*x+1), x) as 2*exp(2*x+1). This requires the chain rule. The step label is 'chain', which is acceptable for the trig terms, but the step performs differentiation on three distinct composite functions (sin, cos, exp) simultaneously. The contract requires one rule per step. Applying the chain rule to three different terms in one step violates the 'one thing' constraint. Additionally, the derivative of exp(2x+1) is not a 'known form' like exp(x), so it requires the chain rule, which is not explicitly labeled as a separate step for that term.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.