Derivative of \( \displaystyle \frac{\sqrt{2} e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}}{6} \)
Problem 2.1402 · hard
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}}{6} \).
- \[ \frac{d}{d x} \frac{\sqrt{2} e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}}{6} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\sqrt{2} \frac{d}{d x} e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}}{6} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sqrt{2} \left(e^{3 x - 3} \frac{d}{d x} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)} + \cos{\left(3 x - 3 + \frac{\pi}{4} \right)} \frac{d}{d x} e^{3 x - 3}\right)}{6} \]productApply the product rule.✓ Proved
- \[ = \frac{\sqrt{2} \left(e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)} \frac{d}{d x} \left(3 x - 3\right) + e^{3 x - 3} \frac{d}{d x} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}\right)}{6} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = \frac{\sqrt{2} \left(- e^{3 x - 3} \sin{\left(3 x - 3 + \frac{\pi}{4} \right)} \frac{d}{d x} \left(3 x - 3 + \frac{\pi}{4}\right) + 3 e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}\right)}{6} \]exponentialDifferentiate the exponential function.✓ Proved
- \[ = \frac{\sqrt{2} \left(- 3 e^{3 x - 3} \sin{\left(3 x - 3 + \frac{\pi}{4} \right)} + 3 e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}\right)}{6} \]chain algebraApply the chain rule to the cosine term. Distribute and simplify the terms.✓ Proved
- \[ = \frac{\sqrt{2} \left(- e^{3 x - 3} \sin{\left(3 x - 3 + \frac{\pi}{4} \right)} + e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}\right)}{2} \]constant-multiple simplifyFactor out the constant 3. Simplify the constant coefficient.✓ Proved
- \[ = \frac{\sqrt{2} \left(- \sin{\left(3 x - 3 + \frac{\pi}{4} \right)} + \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}\right) e^{3 x - 3}}{2} \]algebraFactor out the exponential term.✓ Proved
- \[ = - e^{3 x - 3} \sin{\left(3 x - 3 \right)} \]algebra algebra simplify simplifyExpand the cosine of a sum. Substitute cos(pi/4) and sin(pi/4). Combine like terms. Final simplification.✓ 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 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 14 | ✓ 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) — Step 5 applies two rules at once: it uses the exponential rule to differentiate exp(3*x-3) and simultaneously applies the chain rule to the cosine term. The label "exponential" does not account for the chain rule applied to the cosine, violating the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: fail (error) — Step 5 is labeled 'exponential' but performs two distinct operations: it evaluates the derivative of the inner function (3*x - 3) to 3 (which is a 'derivative' step) and substitutes the derivative of the outer exponential function (which is 'exponential'). Applying two rules in one step violates the contract. Additionally, Step 4 is labeled 'chain' but only partially applies the chain rule to the exponential term while leaving the cosine term untouched, which is a valid single-step application, but Step 5's double action is the primary defect.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 is labeled 'exponential' but performs two distinct operations: it evaluates the derivative of the inner function (3*x - 3) to 3 (which is a 'derivative' step) and substitutes the derivative of the outer exponential function (which is 'exponential'). Applying two rules in one step violates the contract. Additionally, Step 4 is labeled 'chain' but only partially applies the chain rule to the exponential term while leaving the cosine term untouched, which is a valid single-step application, but Step 5's double action is the primary defect.gpt-oss:20b: fail (error) 2026-10-03 — Step 5 applies two rules at once: it uses the exponential rule to differentiate exp(3*x-3) and simultaneously applies the chain rule to the cosine term. The label "exponential" does not account for the chain rule applied to the cosine, violating the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and trigonometric identities to reach the stated answer. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-03 — Step 4 incorrectly labels the application of the exponential rule as a chain rule; the derivative of exp(3*x-3) is exp(3*x-3)*Derivative(3*x-3,x), which is an exponential rule, not a chain rule. Additionally, step 5 applies the exponential rule to exp(3*x-3) but also implicitly uses the chain rule for the inner linear function; labeling it solely as "exponential" is incomplete and misleading. These mislabelings violate the one‑rule‑per‑step requirement.
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.