Differentiate the function $ \sec \left\{ {\tan \left( {\sqrt x } \right)} \right\} $ with respect to x.

Hint: In order to solve this question we will use the chain rule according to the chain rule we will first differentiate the main function given to us then we will differentiate the function that is present the function that is present inside the curly bracket and put it in multiplication of main function’s derivative, then we will differentiate the function that is present inside the small bracket and do the same process.

Complete step by step solution:
For solving this question we will take the help of the chain rule so for this we should know the chain rule. So according to the chain rule if the function’s function is present and we are differentiating it we will first put the derivative of function then we put the derivative of side function in multiplication with the derivative of the main function.
According to the mathematical term:
 $ \dfrac{{df\left\{ {g\left( x \right)} \right\}}}{{dx}} = f'\left\{ {g\left( x \right)} \right\}g'\left( x \right) $
Now for this question the derivative of the function:
 $ \sec \left\{ {\tan \left( {\sqrt x } \right)} \right\} $
After differentiating it the differentiation of $ \sec \left\{ {\tan \left( {\sqrt x } \right)} \right\} $ will be $ \sec \left\{ {\tan \left( {\sqrt x } \right)} \right\}.\tan \left\{ {\tan \left( {\sqrt x } \right)} \right\} $
As we know that the derivative of $ \sec x $ is $ \sec x.\tan x $
Now as the side function is present here that is $ \tan \sqrt x $ so the derivative of this function will be $ {\sec ^2}\sqrt x $ so the according to the chain rule we will put it in multiplication of the derivative of the main function.
At last there is one more side to side chain that is $ \sqrt x $ so the derivative of this function will be:
 $ \dfrac{1}{{2\sqrt x }} $ so this will also present in the multiplication of the derivative of the main function.
Hence the final answer will be:
 $ \sec \left\{ {\tan \left( {\sqrt x } \right)} \right\}.\tan \left\{ {\tan \left( {\sqrt x } \right)} \right\}.{\sec ^2}\sqrt x .\dfrac{1}{{2\sqrt x }} $
So this will be the final answer.
So, the correct answer is “ $ \sec \left\{ {\tan \left( {\sqrt x } \right)} \right\}.\tan \left\{ {\tan \left( {\sqrt x } \right)} \right\}.{\sec ^2}\sqrt x .\dfrac{1}{{2\sqrt x }} $ ”.

Note: While solving these types of question there is a shortcut method which is we will start with the derivative of the main function then we will put the derivative if the side function in its multiplication and then if there is any more such side chain is present we will put it in its multiplication and so on.