GATE 2025 DA Q.14: solution and answer

GATE 2025 · DAQ.14 · 1 mark · MCQ

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

(A)
$0$
(B)
$1$
(C)
$\frac{1}{10 !}$
(D)
$\frac{2}{10 !}$

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Answer and solution

ANSWER

$f^{(10)} (0) = 0$ , which is option (A).

Official answer: ASource: Official answer key, GATE 2025 DA (IIT Roorkee)

Recognise $f (x) = \frac{e ^{x} - e ^{- x}}{2} = sinh x$ .
Differentiate once: $f^{'} (x) = \frac{e ^{x} + e ^{- x}}{2}$ . Differentiate again: $f^{''} (x) = \frac{e ^{x} - e ^{- x}}{2} = f (x)$ .
Because $f^{''} = f$ , every second derivative returns the function: $f^{(2 k)} (x) = f (x)$ for all $k \geq 0$ . Taking $k = 5$ gives $f^{(10)} (x) = f (x)$ .
Evaluate at $0$ : $f^{(10)} (0) = f (0) = \frac{e ^{0} - e ^{- 0}}{2} = \frac{1 - 1}{2} = 0$ .
Check with the Maclaurin series. Using $e^{x} = \sum_{n \geq 0} \frac{x ^{n}}{n !}$ and $e^{- x} = \sum_{n \geq 0} \frac{( - 1 ) ^{n} x ^{n}}{n !}$ , we get $f (x) = \frac{1}{2} \sum_{n \geq 0} \frac{( 1 - ( - 1 ) ^{n} ) x ^{n}}{n !} = x + \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} + \dots$ . Only odd powers appear, so the coefficient of $x^{10}$ is $0$ .
Taylor's relation says $f^{(10)} (0) = 10! \times (coefficient of x^{10}) = 10! \times 0 = 0$ , which agrees with step 4. The answer is option (A).

KEY IDEA

$f (x) = sinh x$ is an odd function, so its Maclaurin series has only odd powers. Every even-order derivative at $0$ is therefore $0$ . Equivalently, $f^{''} = f$ and $f (0) = 0$ .

WATCH OUT FOR

Using $f^{(n)} (0) = n! \times \frac{1}{n !}$ for every $n$ and picking $1$ , or confusing the coefficient $\frac{1}{10 !}$ with the derivative. The even-power coefficients of $sinh x$ are zero. Another slip is treating $f^{(10)}$ like the derivative of $cosh x$ , whose value at $0$ is $1$ .

From the GATE 2025 DA paper. Every question of this paper · Download the full paper · All 2 Taylor series questions, with notes