Derivative of \( \displaystyle - \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \)
Problem 2.654 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{5 x} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} \right)} + \cos{\left(5 x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{5 x}\right)}{10} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(5 e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)} + e^{5 x} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{10} \]derivativeDifferentiate the exponential term.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- 5 e^{5 x} \sin{\left(5 x + \frac{\pi}{4} \right)} + 5 e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{10} \]chain algebraApply the chain rule to the cosine term. Distribute the negative sign inside the parentheses.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- e^{5 x} \sin{\left(5 x + \frac{\pi}{4} \right)} + e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{2} \]constant-multiple simplifyFactor out the common constant 5. Simplify the coefficient.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- \sin{\left(5 x + \frac{\pi}{4} \right)} + \cos{\left(5 x + \frac{\pi}{4} \right)}\right) e^{5 x}}{2} \]algebraFactor out the exponential term.✓ Proved
- \[ = e^{5 x} \sin{\left(5 x \right)} \]algebra algebra algebra simplify algebra simplifyExpand the cosine and sine of the sum using angle addition identities. Substitute cos(pi/4) = sin(pi/4) = sqrt(2)/2. Distribute the negative sign through the second parentheses. Combine like terms inside the parentheses. Multiply the constants and handle the sign. Simplify the final expression.✓ Proved
Answer \( e^{5 x} \sin{\left(5 x \right)} \)
✓ 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 |
| 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 |
| 15 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and trigonometric identities. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies differentiation rules and trigonometric identities. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules and trigonometric identities. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.