Derivative of \( \displaystyle - \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{6} \)
Problem 2.144 · hard
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{6} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{6}\right) \]constantStart with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{6} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{3 x - 3} \frac{d}{d x} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} + \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} \frac{d}{d x} e^{3 x - 3}\right)}{6} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(3 e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} - 3 e^{3 x - 3} \cos{\left(- 3 x + \frac{\pi}{4} + 3 \right)}\right)}{6} \]chain algebraApply the chain rule to both terms. Simplify the expression by distributing the negative sign.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} - e^{3 x - 3} \cos{\left(- 3 x + \frac{\pi}{4} + 3 \right)}\right)}{2} \]simplify simplifyFactor out the common constant 3. Simplify the fraction 3/6 to 1/2.✓ Proved
- \[ = - \frac{\sqrt{2} \left(\sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} - \cos{\left(- 3 x + \frac{\pi}{4} + 3 \right)}\right) e^{3 x - 3}}{2} \]simplifyFactor out the common exponential term.✓ Proved
Answer \( e^{3 x - 3} \sin{\left(3 x - 3 \right)} \)
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 |
| 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) — The final derivative expression is not algebraically equivalent to the stated answer exp(3*x-3)*sin(3*x-3). The steps produce a different function, indicating an incorrect differentiation or simplification.deepseek-r1:70b: fail (misleading) — Step 1 is mislabeled as 'constant' instead of 'derivative'.qwen3.6:27b-mlx: fail (error) — The solution stops at step 8, which is an intermediate expression, rather than providing the final simplified answer stated in the problem description (exp(3*x - 3)*sin(3*x - 3)). The steps provided do not reach the stated answer.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — The solution stops at step 8, which is an intermediate expression, rather than providing the final simplified answer stated in the problem description (exp(3*x - 3)*sin(3*x - 3)). The steps provided do not reach the stated answer.gpt-oss:20b: fail (error) 2026-09-20 — The final derivative expression is not algebraically equivalent to the stated answer exp(3*x-3)*sin(3*x-3). The steps produce a different function, indicating an incorrect differentiation or simplification.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies the chain rule to both terms of the product in a single line, violating the rule that each step must change only one thing. It should be split into two separate chain-rule applications.qwen3.6:27b-mlx: fail (error) 2026-09-19 — The solution stops at step 8, which is an intermediate expression, not the final simplified answer provided in the 'Stated answer'. The steps do not show the trigonometric simplification required to transform the result into exp(3*x - 3)*sin(3*x - 3).deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 1 is mislabeled as 'constant' instead of 'derivative'.gpt-oss:20b: fail (error) 2026-09-19 — Step 4 applies the chain rule to two separate terms in a single line, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications. Although the final expression is not in the simplest form (it can be further simplified to the stated answer using trigonometric identities), the steps provided are mathematically correct and adhere to the contract.deepseek-r1:70b: fail (error) 2026-09-19 — The final answer incorrectly handles the sine function's argument, failing to account for the negative coefficient, leading to an erroneous result.gpt-oss:20b: fail (style) 2026-09-19 — Step 4 applies the chain rule twice (to both derivative terms) in a single line, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (error) 2026-09-18 — The solution stops at step 8, which is an intermediate expression, not the final simplified answer provided in the 'Stated answer'. The steps do not show the trigonometric simplification required to reach exp(3*x - 3)*sin(3*x - 3).deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 4 applies the chain rule twice in a single step, violating the rule that each step must change only one thing. Additionally, the solution stops short of simplifying the result to the expected form exp(3*x-3)*sin(3*x-3).gpt-oss:20b: fail 2026-09-17 — Step 5’s note claims a negative sign is distributed, but the step merely rewrites the expression; this misleads a student about the algebraic manipulation performed.deepseek-r1:70b: pass 2026-09-17
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.