91Ó°ÊÓ

For the function \(y=\tan x\), we need to differentiate it with respect to x: $$y^{\prime} = \frac{d}{dx}(\tan x) = \sec^2{x}$$ The derivative of the tangent function, \(\tan x\), is \(\sec^2{x}\).

Now, we need to differentiate the first derivative, \(y^{\prime} = \sec^2{x}\), with respect to x: $$y^{\prime\prime} = \frac{d^2}{dx^2}(\sec^2 x)$$ To find this derivative, we can use the chain rule, since \(\sec^2{x}\) is a composition of functions. Recall that the chain rule states that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function: $$y^{\prime\prime} = (\frac{d}{dx}(\sec x))^2 \frac{d}{dx}x$$ The derivative of the secant function, \(\sec x\), is \(\sec{x}\tan{x}\) and the derivative of x is 1. Therefore, we have: $$y^{\prime\prime} = (\sec{x}\tan{x})^2(1) = \sec^2{x}\tan^2{x}$$ The second derivative of the function \(y=\tan x\) is \(y^{\prime \prime} = \sec^2{x}\tan^2{x}\).