Question-14

calculus
functions
derivatives
DA-2025

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

We have:

\[ \begin{aligned} f^{( 1)}( x) =f^{\prime }( x) & =\frac{e^{x} +e^{-x}}{2}\\ f^{( 2)}( x) =f^{\prime \prime }( x) & =\frac{e^{x} -e^{-x}}{2} \end{aligned} \]

We see that \(f^{( 2n)}( x) =f( x)\). Therefore, \(f^{( 10)}( 0) =f( 0) =0\).