Derivative of \( \displaystyle e^{- 2 x} \sin{\left(2 x \right)} \)
Problem 2.604 · hard
Differentiate \( \displaystyle f(x) = e^{- 2 x} \sin{\left(2 x \right)} \).
- \[ \frac{d}{d x} e^{- 2 x} \sin{\left(2 x \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \sin{\left(2 x \right)} \frac{d}{d x} e^{- 2 x} + e^{- 2 x} \frac{d}{d x} \sin{\left(2 x \right)} \]productApply the product rule.✓ Proved
- \[ = - 2 e^{- 2 x} \sin{\left(2 x \right)} + e^{- 2 x} \frac{d}{d x} \sin{\left(2 x \right)} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = - 2 e^{- 2 x} \sin{\left(2 x \right)} + 2 e^{- 2 x} \cos{\left(2 x \right)} \]derivative algebraDifferentiate the second part of the product using the chain rule. Rearrange the terms.✓ Proved
- \[ = 2 \left(- \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- 2 x} \]simplifyFactor out the common term.✓ Proved
Answer \( \left(- 2 \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right) e^{- 2 x} \)
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 |
| 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 is labeled 'derivative' but applies the chain rule to differentiate sin(2*x); the label should be 'chain'. Additionally, Step 3 is labeled 'derivative' but only differentiates one part of the product while leaving the other derivative term untouched, which is a partial application not covered by the 'derivative' rule (which implies unfolding d/dx on a known form like exp or sin directly, not a mixed product term).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 4 is labeled 'derivative' but applies the chain rule to differentiate sin(2*x); the label should be 'chain'. Additionally, Step 3 is labeled 'derivative' but only differentiates one part of the product while leaving the other derivative term untouched, which is a partial application not covered by the 'derivative' rule (which implies unfolding d/dx on a known form like exp or sin directly, not a mixed product term).gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 applies the chain rule to differentiate exp(-2*x) but is labeled 'derivative'; the label 'chain' is required when differentiating a composite function. Step 4 is labeled 'derivative' but explicitly notes the use of the chain rule, creating inconsistency and mislabeling.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.