# What is the derivative of #y= sec^3 x+ tan^2 x sec x#?

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The derivative of y = sec^3(x) + tan^2(x)sec(x) is 3sec^2(x)tan(x)sec(x) + 2tan(x)sec^2(x)sec(x) = 3sec^3(x)tan(x) + 2tan(x)sec^3(x) = 5sec^3(x)tan(x).

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To find the derivative of ( y = \sec^3(x) + \tan^2(x) \sec(x) ), we'll use the sum rule and the chain rule.

First, let's find the derivative of ( \sec^3(x) ): [ \frac{d}{dx} (\sec^3(x)) = 3\sec^2(x) \sec(x) \tan(x) ]

Now, let's find the derivative of ( \tan^2(x) \sec(x) ): [ \frac{d}{dx} (\tan^2(x) \sec(x)) = \sec(x) (2\tan(x)\sec(x)\tan(x) + \tan^2(x) \sec(x)) ]

Now, we add the two derivatives together to get the final derivative: [ \frac{d}{dx} (y) = 3\sec^2(x) \sec(x) \tan(x) + \sec(x) (2\tan(x)\sec(x)\tan(x) + \tan^2(x) \sec(x)) ]

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