find \\( \\frac { d y } { d x } \\).\n\\( y = 2 \\sec x \\tan x \\)\n\\( \\frac { d y } { d x } = \\)

Explanation:

Step1: Apply the product rule

The product rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). Let (u = 2\sec x) and (v=\tan x). First, find (u^\prime) and (v^\prime). The derivative of (\sec x) is (\sec x\tan x), so (u^\prime=2\sec x\tan x). The derivative of (\tan x) is (\sec^{2}x), so (v^\prime=\sec^{2}x).

Step2: Substitute into the product rule formula

[ \begin{align*} \frac{dy}{dx}&=(2\sec x\tan x)\cdot\tan x+2\sec x\cdot\sec^{2}x\ &=2\sec x\tan^{2}x + 2\sec^{3}x\ &=2\sec x(\tan^{2}x+\sec^{2}x) \end{align*} ] Using the identity (\tan^{2}x + 1=\sec^{2}x), then (\tan^{2}x=\sec^{2}x - 1). [ \begin{align*} \frac{dy}{dx}&=2\sec x((\sec^{2}x - 1)+\sec^{2}x)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(1 + \tan^{2}x+\sec^{2}x-\tan^{2}x - 1+\sec^{2}x)\ &=2\sec x(2\sec^{2}x-1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1 + \tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+(1+\tan^{2}x)-1)\ &=2\sec x(2\sec^{2}x - 1)\ &=2\sec x(\sec^{2}x+\sec^{2}x - 1)\ &=2\sec x(\