Derivative of \( \displaystyle - \frac{\sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \)

Problem 2.4 · hard Video

Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \).

\[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{\sqrt{2} \frac{d}{d x} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \]
constant-multipleFactor out the constant term.✓ Proved
\[ = - \frac{\sqrt{2} \left(e^{x + 1} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{x + 1}\right)}{2} \]
productApply the product rule.✓ Proved
\[ = - \frac{\sqrt{2} \left(e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)} + e^{x + 1} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]
derivativeCompute the derivative of exp(x + 1).✓ Proved
\[ = - \frac{\sqrt{2} \left(- e^{x + 1} \sin{\left(x + \frac{\pi}{4} + 1 \right)} + e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]
derivative algebraCompute the derivative of cos(x + pi/4 + 1). Simplify the expression inside the parentheses.✓ Proved
\[ = - \frac{\sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right) e^{x + 1}}{2} \]
algebraFactor out exp(x + 1).✓ Proved
\[ = e^{x + 1} \sin{\left(x + 1 \right)} \]
simplify algebra algebraUse the trigonometric identity cos(theta) - sin(theta) = sqrt(2)cos(theta + pi/4). Combine the pi/4 terms. Multiply the constants (sqrt(2)/2 * sqrt(2) = 1).✓ Proved

Answer \( e^{x + 1} \sin{\left(x + 1 \right)} \)

Top: the graph of f = -sqrt(2)·e^(x + 1)·cos(x + pi/4 + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sqrt(2)·e^(x + 1)·sin(x + pi/4 + 1)/2 - sqrt(2)·e^(x + 1)·cos(x + pi/4 + 1)/2.

✓ 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

deepseek-r1:70b: pass
gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and trigonometric identities to reach the stated answer. Each step isolates a single rule application, and the labels are appropriate.

Senior review claude-sonnet-5-5, 2026-10-03: pass — The math is correct: cos θ − sin θ = √2·cos(θ + π/4) with θ = x + 1 + π/4, so the result is −e^(x+1)·cos(x + 1 + π/2) = e^(x+1)·sin(x + 1), matching the stated answer. Every line was proved by SymPy and the rule labels are acceptable. The only blemish is that the last line is not rewritten into the stated form, which is not a defect.

Every verdict on record (12)

claude-sonnet-5-5: pass 2026-10-03 — The math is correct: cos θ − sin θ = √2·cos(θ + π/4) with θ = x + 1 + π/4, so the result is −e^(x+1)·cos(x + 1 + π/2) = e^(x+1)·sin(x + 1), matching the stated answer. Every line was proved by SymPy and the rule labels are acceptable. The only blemish is that the last line is not rewritten into the stated form, which is not a defect.
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and trigonometric identities to reach the stated answer. Each step isolates a single rule application, and the labels are appropriate.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and trigonometric identities to reach the stated answer. Each step isolates a single operation, and the labels accurately reflect the transformations applied.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and trigonometric identities to reach the stated answer. Each step isolates a single operation, and the labels accurately reflect the rules applied.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: inconclusive 2026-09-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 5 incorrectly computes the derivative of cos(x + pi/4 + 1) as -sin(x + pi/4 + 1), omitting the factor of 1 from the chain rule (derivative of t
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-16

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.