Home›Calculus 1›Chain rule›Problem 2.207

Derivative of \( \displaystyle - \frac{5 \sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \)

Problem 2.207 · hard Video

Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \).

\[ \frac{d}{d x} \left(- \frac{5 \sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{5 \sqrt{2} \frac{d}{d x} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{5 \sqrt{2} \left(e^{x + 1} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{x + 1}\right)}{2} \]
productApply the product rule to the remaining terms.✓ Proved
\[ = - \frac{5 \sqrt{2} \left(- e^{x + 1} \sin{\left(x + \frac{\pi}{4} + 1 \right)} + e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]
derivativeDifferentiate the exponential and cosine terms.✓ Proved
\[ = - \frac{5 \sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right) e^{x + 1}}{2} \]
algebraFactor out the common exponential term.✓ Proved
\[ = 5 e^{x + 1} \sin{\left(x + 1 \right)} \]
simplify simplifyUse the trigonometric identity cos(A) - sin(A) = sqrt(2)cos(A + pi/4). Simplify the expression by combining the sqrt(2) terms.✓ Proved

Answer \( 5 e^{x + 1} \sin{\left(x + 1 \right)} \)

Top: the graph of f = -5sqrt(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' = 5sqrt(2)·e^(x + 1)·sin(x + pi/4 + 1)/2 - 5sqrt(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
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final derivative computed in the solution is \u22125 e^{x+1}\cos(x+π/2+1), which is algebraically correct, but it does not match the stated answer 5 e
deepseek-r1:70b: fail (error) — The first step lacks a rule label, which is required by the contract.
qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple, product, and derivative rules, followed by valid trigonometric simplifications to reach the stated answer. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.

Senior review claude-sonnet-5-5, 2026-10-06: pass — Each step is a valid equality with an acceptable rule label. The identity cos A - sin A = √2 cos(A+π/4) is correct, and -5 e^{x+1} cos(x+π/2+1) = 5 e^{x+1} sin(x+1), which is the stated answer.

gpt-oss:20b: dismiss — The objection says the final line -5 e^{x+1} cos(x+π/2+1) does not match 5 e^{x+1} sin(x+1), but cos(θ+π/2) = -sin θ makes them equal, and SymPy proved the final line equals the stated answer.

Every verdict on record (13)

claude-sonnet-5-5: pass 2026-10-06 — Each step is a valid equality with an acceptable rule label. The identity cos A - sin A = √2 cos(A+π/4) is correct, and -5 e^{x+1} cos(x+π/2+1) = 5 e^{x+1} sin(x+1), which is the stated answer.
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple, product, and derivative rules, followed by valid trigonometric simplifications to reach the stated answer. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
gpt-oss:20b: inconclusive 2026-09-20 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final derivative computed in the solution is \u22125 e^{x+1}\cos(x+π/2+1), which is algebraically correct, but it does not match the stated answer 5 e
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple, product, and derivative rules, followed by valid algebraic and trigonometric simplifications to reach the stated answer.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple, product, and derivative rules, followed by valid trigonometric simplifications to reach the stated answer.
deepseek-r1:70b: fail (error) 2026-09-19 — The first step lacks a rule label, which is required by the contract.
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies the constant-multiple, product, and derivative rules, followed by valid algebraic and trigonometric simplifications to reach the stated answer.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: fail 2026-09-17 — Step 4 is mislabeled as a derivative step—it merely repeats the previous expression without applying any rule, which could mislead a student about what was actually done.
deepseek-r1:70b: fail 2026-09-17 — The final answer has an incorrect positive sign; the correct derivative should be negative.

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.