Question

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

Explanation:

Step1: Apply the product rule

The product rule states that if \(y = uv\), then \(\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\). Here \(u = 2\sec x\) and \(v=\tan x\).
\(\frac{dy}{dx}=2\sec x\frac{d}{dx}(\tan x)+2\tan x\frac{d}{dx}(\sec x)\)

Step2: Differentiate \(\tan x\) and \(\sec x\)

We know that \(\frac{d}{dx}(\tan x)=\sec^{2}x\) and \(\frac{d}{dx}(\sec x)=\sec x\tan x\).
Substituting these values:

$$
LATEXBLOCK0
$$

Using the identity \(\sec^{2}x = 1+\tan^{2}x\), we can also rewrite it as \(2\sec x(1 + 2\tan^{2}x)\) or \(2\sec x(2\sec^{2}x-1)\)

Answer:

\(2\sec^{3}x + 2\sec x\tan^{2}x\)