Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2232

Derivatives and integrals of hyperbolic functions

Problem 2.2232 · easy

Differentiate \( \displaystyle f(x) = - \tanh{\left(x + 2 \right)} \).

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

Answer \( f'(x) = \tanh^{2}{\left(x + 2 \right)} - 1 \)

✓ 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 applies the chain rule and the identity sech^2(u) = 1 - tanh^2(u) to derive the result. The stated answer is mathematically equivalent to the standard derivative form.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the chain rule and the identity sech^2(u) = 1 - tanh^2(u) to derive the result. The stated answer is mathematically equivalent to the standard derivative form.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the chain rule and the identity sech^2(u) = tanh^2(u) - 1. The final form is mathematically equivalent to the standard derivative -sech^2(x+2).
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the chain rule and the identity sech^2(u) = 1 - tanh^2(u) to derive the stated answer.
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 2026-10-17 with SymPy 1.14.0.