Differentiate the following functions with respect to x

sec ( $t a n \sqrt{x}$ )

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Solution

Ley y = sec ( $t a n \sqrt{x}$ )

Differentiate both sides w.r.t. x, we get

$\frac{d y}{d x} = \frac{d}{d x} (s e c (t a n \sqrt{x})) = s e c (t a n \sqrt{x}) t a n (t a n \sqrt{x}) \frac{d}{d x} (t a n \sqrt{x})$ (By chain rule)

= $s e c (t a n \sqrt{x}) t a n (t a n \sqrt{x}) s e c^{2} \sqrt{x} \frac{d}{d x} (\sqrt{x})$

= $s e c (t a n \sqrt{x}) t a n (t a n \sqrt{x}) (s e c^{2} \sqrt{x}) (\frac{1}{2 \sqrt{x}})$

= $\frac{1}{2 \sqrt{x}} s e c (t a n \sqrt{x}) t a n (t a n \sqrt{x}) (s e c^{2} \sqrt{x})$

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