Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2356

Derivatives and integrals of hyperbolic functions

Problem 2.2356 · easy

Differentiate \( \displaystyle f(x) = \cosh{\left(2 x \right)} \).

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

Answer \( f'(x) = 2 \sinh{\left(2 x \right)} \)

✓ 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 identifies and applies the chain rule for the derivative of cosh. The stated formulas in the first line are correct, and the final result matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies and applies the chain rule for the derivative of cosh. The stated formulas in the first line are correct, and the final result matches the stated answer.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies and applies the chain rule for the derivative of the hyperbolic cosine function. The stated differentiation rules are accurate.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the chain rule and the standard derivative formula for hyperbolic cosine. The setup and conclusion are sound.
gpt-oss:20b: pass 2026-10-15

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.