Derivative of \( \displaystyle \frac{\sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \)
Problem 2.426 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\sqrt{2} \frac{d}{d x} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sqrt{2} \left(e^{x - 3} \frac{d}{d x} \cos{\left(x - 3 + \frac{\pi}{4} \right)} + \cos{\left(x - 3 + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x - 3}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = \frac{\sqrt{2} \left(e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)} + e^{x - 3} \frac{d}{d x} \cos{\left(x - 3 + \frac{\pi}{4} \right)}\right)}{2} \]derivativeDifferentiate the exponential term.✓ Proved
- \[ = \frac{\sqrt{2} \left(- e^{x - 3} \sin{\left(x - 3 + \frac{\pi}{4} \right)} \frac{d}{d x} \left(x - 3 + \frac{\pi}{4}\right) + e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}\right)}{2} \]trigDifferentiate the cosine term using the chain rule.✓ Proved
- \[ = \frac{\sqrt{2} \left(- e^{x - 3} \sin{\left(x - 3 + \frac{\pi}{4} \right)} + e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}\right)}{2} \]derivative algebraDifferentiate the inner linear function. Distribute and simplify the signs.✓ Proved
- \[ = \frac{\sqrt{2} \left(- \sin{\left(x - 3 + \frac{\pi}{4} \right)} + \cos{\left(x - 3 + \frac{\pi}{4} \right)}\right) e^{x - 3}}{2} \]algebraFactor out the common exponential term.✓ Proved
- \[ = - e^{x - 3} \sin{\left(x - 3 \right)} \]simplify simplifyUse the trigonometric identity cos(A) - sin(A) = sqrt(2)cos(A + pi/4). Simplify the coefficients.✓ Proved
Answer \( - e^{x - 3} \sin{\left(x - 3 \right)} \)
✓ 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 |
| 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 |
| 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 (style) — Step 5 applies both the trig derivative rule and the chain rule in one line, but it is labeled only as "trig". This violates the rule that each step must change exactly one thing and use a single appropriate label.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and trigonometric identities. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Senior review claude-sonnet-5-5, 2026-10-08: pass — Every line was proved equivalent by SymPy, including the final line against the stated answer. The rule labels are acceptable, and the step 9 note quotes the identity loosely but the line itself is correct.
gpt-oss:20b: dismiss — Differentiating cos(u) to -sin(u)*u' is a single chain-rule application, and 'trig' is a legitimate label for it (two labels fit, either acceptable); evaluating the inner derivative is part of that step.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — Every line was proved equivalent by SymPy, including the final line against the stated answer. The rule labels are acceptable, and the step 9 note quotes the identity loosely but the line itself is correct.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules and trigonometric identities. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (style) 2026-09-21 — Step 5 applies both the trig derivative rule and the chain rule in one line, but it is labeled only as "trig". This violates the rule that each step must change exactly one thing and use a single appropriate label.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 is labeled 'trig' but applies the chain rule to differentiate the cosine term; the label should be 'chain'. Step 9 uses a trigonometric identity but is labeled 'simplify', which is acceptable, but the primary defect is the mislabeling of the chain rule application in step 5.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 9 applies a trigonometric identity that is mathematically incorrect; cos(A) - sin(A) equals sqrt(2)cos(A + pi/4), not sqrt(2)cos(A + pi/2). This leads to an incorrect final answer.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies both the trigonometric derivative rule and the chain rule in a single step, violating the one‑rule‑per‑step rule. It should be split into two separate steps: first differentiate the cosine (trig) and then apply the chain rule to the inner linear function.
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.