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$Let\quad y={ e }^{ 2\sin ^{ -1 }{ x } }$

Now , differentiating with respect to $x$ , we get

$\\ { y }'=\frac { 2y }{ \sqrt { 1-{ x }^{ 2 } } } \\ \left( \sqrt { 1-{ x }^{ 2 } } \right) { y }'=2y$

Again differentiating with respect to $x$ and simplifying the euation, we get

$\left( 1-{ x }^{ 2 } \right) y''-xy'=4y$ ------------equation $1$

Here, we use the Leibnitz's rule of finding $nth$ derivative of product of two functions:

${ \left( uv \right) }_{ n }=\left( \begin{matrix} n \\ 0 \end{matrix} \right) { u }{ v }_{ n }+\left( \begin{matrix} n \\ 1 \end{matrix} \right) { u }_{ 1 }{ v }_{ n-1 }+\left( \begin{matrix} n \\ 2 \end{matrix} \right) { u }_{ 2 }{ v }_{ n-2 }+......+\left( \begin{matrix} n \\ n \end{matrix} \right) { u }_{ n }{ v }$

where ${ x }_{ n }$ is the $nth$ derivative of $x$ and $\left( \begin{matrix} a \\ b \end{matrix} \right) =\frac { a! }{ b!(a-b)! }$

Now, applying Leibnitz's rule to equation $1$ , we get

$\left( 1-{ x }^{ 2 } \right) { y }_{ n+2 }-2nx{ y }_{ n+1 }-n\left( n-1 \right) { y }_{ n }-x{ y }_{ n+1 }-n{ y }_{ n }=4{ y }_{ n }$

On simplifying, we get

$\left( 1-{ x }^{ 2 } \right) { y }_{ n+2 }-(2n+1)x{ y }_{ n+1 }-(4+{ n }^{ 2 }){ y }_{ n }=0$ -------------equation $2$

where ${ y }_{ n }$ is the $nth$ derivative of $y$ . Now, substituting value of $x$ as $0$ in equation $2$ , we get

${ y }_{ n+2 }=\left( 4+{ n }^{ 2 } \right) { y }_{ n }$ ---------equation $3$

We know that, ${ y }\left( 0 \right) =1$

${ y }_{ 1 }\left( 0 \right) =2$

Using equation $3$ , relation for ${ y }_{ n }\left( 0 \right)$ becomes ,

${ y }_{ n }\left( 0 \right) =2\left( { 2 }^{ 2 }+{ 1 }^{ 2 } \right) \left( { 2 }^{ 2 }+{ 3 }^{ 2 } \right) .....\left( { 2 }^{ 2 }+{ \left( n-2 \right) }^{ 2 } \right) ,\quad n\quad is\quad odd\\ { y }_{ n }\left( 0 \right) ={ 2 }^{ 2 }\left( { 2 }^{ 2 }+{ 2 }^{ 2 } \right) \left( { 2 }^{ 2 }+{ 4 }^{ 2 } \right) .....\left( { 2 }^{ 2 }+{ \left( n-2 \right) }^{ 2 } \right) ,\quad n\quad is\quad even$

Substituting value of $n=10$ , we get ${ y }_{ 10 }\left( 0 \right) =1740800$