Question

$differentiate \: w.r.t. \: \sqrt{ \tan( \sqrt{x} ) }$
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Answer 1

Given :

$\sf \sqrt{ tan( \sqrt{x} ) }$

To find :

• differentiate the given expression with respect to x

Solution :

$\sf \implies \: \dfrac{d(\sqrt{ tan( \sqrt{x} ) })}{dx}$

Using Chain rule we will differentiate.

$\sf \implies \dfrac{1}{2} \times \bigg({ tan( \sqrt{x}}) \bigg)^{ 1 - \frac{1}{2} } \times \dfrac{d({ tan( \sqrt{x} ) })}{dx} \times \dfrac{d({\sqrt{x} })}{dx}$

$\sf \implies \dfrac{1}{2} \times \bigg({ tan( \sqrt{x})} \bigg)^{ - \frac{1}{2} } \times {{ {sec}^{2} ( \sqrt{x} ) }} \times \dfrac{1}{2} {({ \dfrac{1}{\sqrt{x}} })}$

$\sf \implies \dfrac{{sec}^{2} ( \sqrt{x} ) }{4\sqrt{x} \sqrt{tan( \sqrt{x})} }$

Answer :

$\bf \large \dfrac{{sec}^{2} ( \sqrt{x} ) }{4\sqrt{x} \sqrt{tan( \sqrt{x})} }$