Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2445

Derivatives and integrals of hyperbolic functions

Problem 2.2445 · medium

Differentiate \( \displaystyle f(x) = \tanh{\left(\sqrt{x} \right)} \).

(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
\[ \frac{d}{d x} \tanh{\left(\sqrt{x} \right)} = \frac{1}{2 \sqrt{x} \cosh^{2}{\left(\sqrt{x} \right)}} \]
Apply the rule (and the chain rule).✓ Proved

Answer \( f'(x) = \frac{1}{2 \sqrt{x} \cosh^{2}{\left(\sqrt{x} \right)}} \)

✓ 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 applies the chain rule and the derivative of the hyperbolic tangent function. The final answer is mathematically equivalent to the standard form using sech, as sech^2(u) = 1/cosh^2(u).

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the chain rule and the derivative of the hyperbolic tangent function. The final answer is mathematically equivalent to the standard form using sech, as sech^2(u) = 1/cosh^2(u).
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the chain rule application and the derivative of tanh. The final expression is mathematically equivalent to the standard form using sech^2, as sech(u) = 1/cosh(u).
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the chain rule and the standard derivative of tanh. The intermediate step listing derivative formulas is correct, and the final result matches the stated answer.
gpt-oss:20b: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.