Gate DA 2025 | Question - 14 | Engineering Mathematics

Let $f (x) = \frac{e^{x} - e^{- x}}{2}$ , $x \in ℝ$ . Let $f^{(k)} (a)$ denote the k^th derivative of $f$ evaluated at $a$ . What is the value of $f^{(10)} (0)$ ? (Note: ! denotes factorial)

$\frac{1}{10!}$

$\frac{2}{10!}$

Solution:

We are given the function:

$f (x) = \frac{e^{x} - e^{- x}}{2}$

This is the definition of the hyperbolic sine function:

$f (x) = \sinh (x)$

We are asked to find the 10th derivative of $f$ at $x = 0$ , i.e., $f^{(10)}(0)$ .

Key Observations:

The Taylor series expansion of $\sinh(x)$ is:
$\sinh (x) = \sum_{n = 0}^{\infty} \frac{x^{2 n + 1}}{(2 n + 1)!}$

This series only contains odd powers of $x$ , which means:

All even derivatives (like the 2nd, 4th, 6th, ..., 10th) of $\sinh(x)$ at $x = 0$ are zero.

Therefore:

$f^{(10)} (0) = 0$

Answer: (A) 0

Gate DA 2025 | Question - 14 | Engineering Mathematics

Key Observations:

Therefore:

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