Problem

Find $\frac{d y}{d x}$
\[
y=6(\tan x+\sec x)(\tan x-\sec x)
\]

Answer

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Answer

\(\boxed{\frac{d y}{d x} = (\tan x - \sec x)*(6\tan x^2 + 6\tan x\sec x + 6) + (6\tan x + 6\sec x)*(\tan x^2 - \tan x\sec x + 1)}\) is the final answer.

Steps

Step 1 :Given the function \(y=6(\tan x+\sec x)(\tan x-\sec x)\)

Step 2 :We need to find the derivative of this function. We can use the product rule of differentiation which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 3 :The derivative of \(\tan x\) is \(\sec^2 x\) and the derivative of \(\sec x\) is \(\sec x \tan x\).

Step 4 :So, we first need to find the derivatives of \((\tan x + \sec x)\) and \((\tan x - \sec x)\), then apply the product rule.

Step 5 :Applying the product rule, we get \((\tan x - \sec x)*(6\tan x^2 + 6\tan x\sec x + 6) + (6\tan x + 6\sec x)*(\tan x^2 - \tan x\sec x + 1)\)

Step 6 :\(\boxed{\frac{d y}{d x} = (\tan x - \sec x)*(6\tan x^2 + 6\tan x\sec x + 6) + (6\tan x + 6\sec x)*(\tan x^2 - \tan x\sec x + 1)}\) is the final answer.

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