Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2237

Derivatives and integrals of hyperbolic functions

Problem 2.2237 · easy

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

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

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

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	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: fail (error) — The solution claims to apply the listed rules to derive the result, but the listed rules are for single hyperbolic functions, not the product of two. The step skips the necessary application of the product rule and the identity cosh^2(x) + sinh^2(x) = cosh(2x), making the derivation logically incomplete and misleading.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution claims to apply the listed rules to derive the result, but the listed rules are for single hyperbolic functions, not the product of two. The step skips the necessary application of the product rule and the identity cosh^2(x) + sinh^2(x) = cosh(2x), making the derivation logically incomplete and misleading.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the derivative using standard hyperbolic identities or product rule, and the final result is correct.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution lists derivative formulas but fails to apply the product rule to differentiate the product of sinh(x) and cosh(x). It jumps directly to the answer without showing the necessary intermediate step (e.g., using the identity sinh(x)cosh(x) = (1/2)sinh(2x) or applying the product rule explicitly).
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.