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Compute the derivative of a given function

by RoRi

Compute the derivative of

    \[ f(x) = \tan x \sec x. \]

We recall from the previous exercise (Apostol, 4.6 Exercise #25) that (\tan x)' = \sec^2 x and (\sec x)' = \tan x \sec x. Then, using the product rule we have,

    \begin{align*} f'(x) &= (\tan x)' \sec x + \tan x (\sec x)' \\ &= (\sec^2 x) \sec x + \tan x (\tan x \sec x) \\ &= \sec^3 x + \tan^2 x \sec x. \end{align*}

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