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Derivative of an inverse function

Problem 2.1545 · medium

Let \( \displaystyle f(x) = 2 x + \sin{\left(x \right)} \), which is one-to-one. Find \( \displaystyle (f^{-1})'(0) \).

(f⁻¹)′(a) = 1 / f′(f⁻¹(a)). First find the x with f(x) = a; a small integer is worth trying.
Reviewed
\[ \left. 2 x + \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]
f(0) = 0, so f⁻¹(0) = 0.✓ Proved
\[ \frac{d}{d x} \left(2 x + \sin{\left(x \right)}\right) = \cos{\left(x \right)} + 2 \]
Differentiate f.✓ Proved
\[ \frac{1}{\left. \cos{\left(x \right)} + 2 \right|_{\substack{ x=0 }}} = \frac{1}{3} \]
Take the reciprocal of f′ there.✓ Proved

Answer \( (f^{-1})'(0) = \frac{1}{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	inverted f numerically at a ± 1e-10 and took the difference quotient

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the inverse function theorem, identifies the correct pre-image, and computes the derivative accurately.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the inverse function theorem, identifies the correct pre-image, and computes the derivative accurately.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04
gpt-oss:20b: pass 2026-10-04

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 generator:structured/inverse_derivative, checked 2026-10-04 with SymPy 1.14.0.