Derivative of \( \displaystyle e^{\cos{\left(x \right)}} \)
Problem 2.649 · hard
Differentiate \( \displaystyle f(x) = e^{\cos{\left(x \right)}} \).
- \[ \frac{d}{d x} e^{\cos{\left(x \right)}} \]Start with the derivative of the function.✓ Proved
- \[ = e^{\cos{\left(x \right)}} \frac{d}{d x} \cos{\left(x \right)} \]chainApply the chain rule.✓ Proved
- \[ = - e^{\cos{\left(x \right)}} \sin{\left(x \right)} \]trigDifferentiate cos(x) to get -sin(x).✓ Proved
Answer \( - e^{\cos{\left(x \right)}} \sin{\left(x \right)} \)
✓ 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.
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 |
| 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-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.