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Compute the derivative of the given function

by RoRi

Compute the derivative of the function

    \[ f(x) = \sec^2 x + \csc^2 x. \]

First, we simplify the given expression,

    \begin{align*} f(x) = \sec^2 x + \csc^2 x &= \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} \\ &= \frac{\sin^2 x + \cos^2 x}{\cos^2 x \cdot \sin^2 x} \\ &= \frac{1}{\frac{1}{4} \sin^2 (2x)} \\ &= \frac{4}{\sin^2 (2x)}. \end{align*}

Then, we take the derivative,

    \begin{align*} f'(x) &= \frac{-4(2 \cos (2x))(\sin (2x))(2)}{\sin^4 (2x)} \\[8pt] &= - \frac{16 \cos (2x)}{\sin^3 (2x)}. \end{align*}

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