The GATE Grind

GATE 2025 DA – Question 14

Calculus and Optimization · Taylor series, maxima and minima, single-variable optimization · 1 mark · Multiple choice

Let $f(x) = \frac{e^x - e^{-x}}{2}$, $x \in \mathbb{R}$. 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)

  1. 0
  2. 1
  3. $\frac{1}{10!}$
  4. $\frac{2}{10!}$

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Correct answer: (A) 0

Explanation

$f(x) = \sinh x$, whose derivatives alternate between $\cosh x$ and $\sinh x$. The derivative of even order is $\sinh x$ again, so $f^{(10)}(x) = \sinh x$ and $f^{(10)}(0) = \sinh 0 = 0$. In the Taylor series of $\sinh x$ only odd powers appear, so every even derivative at 0 vanishes.