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GATE DA 2025 Q14: 10th derivative of (eˣ − e⁻ˣ)/2 at 0

Question

Let f(x)=ex−e−x2f(x) = \dfrac{e^{x} - e^{-x}}{2}, x∈Rx \in \mathbb{R}. Let f(k)(a)f^{(k)}(a) denote the kthk^{th} derivative of ff evaluated at aa. What is the value of f(10)(0)f^{(10)}(0)? (Note: ! denotes factorial)

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Answer: A (official GATE 2025 final answer key)

Concept used: Derivatives of sinh⁡x\sinh x alternate between cosh⁡x\cosh x and sinh⁡x\sinh x.

First step: Here f(x)=sinh⁡xf(x) = \sinh x, so f′=cosh⁡xf' = \cosh x, f′′=sinh⁡xf'' = \sinh x, and every even-order derivative equals sinh⁡x\sinh x. Evaluate it at 0.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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