Derivative of \( \displaystyle - \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{2} \)
Problem 2.234 · hard
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{3 x + 1} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{3 x + 1}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- e^{3 x + 1} \sin{\left(3 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} \left(3 x + \frac{\pi}{4} + 1\right) + \cos{\left(3 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{3 x + 1}\right)}{2} \]trigDifferentiate the cosine term using the chain rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- 3 e^{3 x + 1} \sin{\left(3 x + \frac{\pi}{4} + 1 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{3 x + 1}\right)}{2} \]derivativeDifferentiate the inner function of the sine term.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- 3 e^{3 x + 1} \sin{\left(3 x + \frac{\pi}{4} + 1 \right)} + 3 e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]derivativeDifferentiate the exponential term.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(- e^{3 x + 1} \sin{\left(3 x + \frac{\pi}{4} + 1 \right)} + e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]algebraDistribute the constant and simplify.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(- \sin{\left(3 x + \frac{\pi}{4} + 1 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}\right) e^{3 x + 1}}{2} \]algebra simplifyFactor out the common exponential term. Final simplified form.✓ Proved
Answer \( 3 e^{3 x + 1} \sin{\left(3 x + 1 \right)} \)
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: fail (error) — The final expression derived in the steps is -3*sqrt(2)/2 * exp(3*x+1)*(cos(3*x+1+pi/4)-sin(3*x+1+pi/4)), which is not algebraically equivalent to the stated answer 3*exp(3*x+1)*sin(3*x+1). The solution therefore contains an error in the final simplification/answer.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — The final answer is mathematically incorrect; the derivative of the given function is not 3*exp(3*x + 1)*sin(3*x + 1). The solution fails to simplify the trigonometric expression in step 9 to match the stated answer, and the stated answer itself is wrong.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — The final answer is mathematically incorrect; the derivative of the given function is not 3*exp(3*x + 1)*sin(3*x + 1). The solution fails to simplify the trigonometric expression in step 9 to match the stated answer, and the stated answer itself is wrong.gpt-oss:20b: fail (error) 2026-09-20 — The final expression derived in the steps is -3*sqrt(2)/2 * exp(3*x+1)*(cos(3*x+1+pi/4)-sin(3*x+1+pi/4)), which is not algebraically equivalent to the stated answer 3*exp(3*x+1)*sin(3*x+1). The solution therefore contains an error in the final simplification/answer.qwen3.6:27b-mlx: fail (error) 2026-09-20 — The final answer provided in the steps does not match the stated answer. The stated answer is 3*exp(3*x + 1)*sin(3*x + 1), but the derived expression in step 9 is -3*sqrt(2)/2 * exp(3*x + 1) * (cos(3*x + 1 + pi/4) - sin(3*x + 1 + pi/4)). While these expressions are mathematically equivalent, the solution fails to perform the necessary trigonometric simplification to reach the stated answer, leaving the result in an unsimplified form that does not match the target.gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies both the chain rule and the derivative of cosine in one move, yet it is labeled only as "trig". This violates the rule that each step must apply a single named rule.qwen3.6:27b-mlx: fail (misleading) 2026-09-19 — The solution fails to simplify the trigonometric expression in the final step, leaving the answer in a form that does not match the stated answer. Specifically, it does not apply the identity -sqrt(2)/2 * (cos(theta) - sin(theta)) = sin(theta - pi/4) to reach the target result.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 4 applies both the chain rule and the trigonometric derivative in one line, but it is labeled only as "trig". According to the contract, each step must apply exactly one rule, so the label should be "chain" (or a separate step for the trigonometric part).qwen3.6:27b-mlx: fail (error) 2026-09-19 — The final answer is mathematically incorrect; the derivative of the given function does not simplify to 3*exp(3*x + 1)*sin(3*x + 1). The solution fails to perform the necessary trigonometric simplification (using angle addition formulas) to reach the stated answer, and the last step shown is not equal to the stated answer.deepseek-r1:70b: fail (error) 2026-09-19 — Step 1 lacks a rule label, violating the contract. Additionally, the final answer differs from the derived expression, which could mislead students.gpt-oss:20b: fail (error) 2026-09-19 — Step 4 applies both the trig derivative rule and the chain rule in a single step, violating the one‑rule‑per‑step requirement.gpt-oss:20b: fail 2026-09-17 — Step 4 applies both the trig derivative and the chain rule but is labeled only as "trig", which misrepresents the operations performed and could mislead a student.deepseek-r1:70b: pass 2026-09-17 — The solution correctly applies differentiation rules and algebraic manipulations without any defects.
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.