Home›Calculus 1›Derivatives of exponential functions›Problem 2.604

Derivative of \( \displaystyle e^{- 2 x} \sin{\left(2 x \right)} \)

Problem 2.604 · hard Video

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 \( \left(- 2 \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right) e^{- 2 x} \)

Top: the graph of f = e^(-2x)·sin(2x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2e^(-2x)·sin(2x) + 2e^(-2x)·cos(2x).

✓ 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
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: pass
qwen3.6:27b-mlx: fail (style) — Step 3 applies the chain rule to differentiate exp(-2*x) but is labeled 'derivative', which is reserved for basic forms; it should be labeled 'chain'. Step 4 is labeled 'derivative' but explicitly notes the use of the chain rule, creating a mismatch between label and note.

Senior review claude-sonnet-5-5, 2026-10-10: pass — The math is correct throughout: product rule, then each factor differentiated, then algebraic rearrangement and factoring, and the final line matches the stated answer. The 'derivative' labels are legitimate, so the label objection is not a defect.

qwen3.6:27b-mlx: dismiss — The contract states that evaluating the derivative of a simple inner function is part of a chain-rule step, and 'derivative' is an allowed label for differentiating a term. Step 4's note mentions the chain rule, but 'derivative' still names a rule the step legitimately applied, and where two labels fit either is acceptable. Both lines were proved equal by SymPy.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — The math is correct throughout: product rule, then each factor differentiated, then algebraic rearrangement and factoring, and the final line matches the stated answer. The 'derivative' labels are legitimate, so the label objection is not a defect.
qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 3 applies the chain rule to differentiate exp(-2*x) but is labeled 'derivative', which is reserved for basic forms; it should be labeled 'chain'. Step 4 is labeled 'derivative' but explicitly notes the use of the chain rule, creating a mismatch between label and note.
gpt-oss:20b: pass 2026-09-26
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 2026-09-21
qwen3.6:27b-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.