Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2311

Derivatives and integrals of hyperbolic functions

Problem 2.2311 · easy

Differentiate \( \displaystyle f(x) = - \operatorname{asinh}{\left(x \right)} \).

(sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
Reviewed
\[ \frac{d}{d x} \left(- \operatorname{asinh}{\left(x \right)}\right) = - \frac{1}{\sqrt{x^{2} + 1}} \]
Apply the rule (and the chain rule).✓ Proved

Answer \( f'(x) = - \frac{1}{\sqrt{x^{2} + 1}} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. 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	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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	a central difference, with sinh and cosh computed from exponentials by mpmath

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies and applies the standard derivative rule for the inverse hyperbolic sine function. The setup is accurate and the conclusion follows directly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies and applies the standard derivative rule for the inverse hyperbolic sine function. The setup is accurate and the conclusion follows directly.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies and applies the standard derivative rule for the inverse hyperbolic sine function, including the chain rule for the negative sign. The setup and conclusion are mathematically sound.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the derivative rule for the inverse hyperbolic sine function and applies it accurately, including the negative sign from the original function.
gpt-oss:20b: pass 2026-10-14

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/hyperbolic_calculus, checked 2026-10-18 with SymPy 1.14.0.