The derivative of $f( an x)$ w.r.t. $g(\sec x)$ at $x=\frac{\pi}{4}$, where $f^{\prime}(1)=2$ and $g^{\prime}(\sqrt{2})=4$ is

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The derivative of $f(	an x)$ w.r.t. $g(\sec x)$ at $x=\frac{\pi}{4}$, where $f^{\prime}(1)=2$ and $g^{\prime}(\sqrt{2})=4$ is

Aicharya · Accepted Answer

$\begin{array}{l} \frac{\frac{d}{d x} f(	an x)}{\frac{d}{d x} g(\sec x)}=\frac{f^{\prime}(	an x) \cdot \sec ^{2} x}{g^{\prime}(\sec x) \cdot \sec x 	an x} \ 	ext { At } x=\frac{\pi}{4} 	ext { we get } \ =\frac{f^{\prime}\left(	an \frac{\pi}{4}ight)\left(\sec ^{2} \frac{\pi}{4}ight)}{g^{\prime}\left(\sec \frac{\pi}{4}ight) \sec \frac{\pi}{4} 	an \frac{\pi}{4}}=\frac{f^{\prime}(1)(2)}{g^{\prime}(\sqrt{2})(\sqrt{2})(1)}=\frac{2 	imes 2}{4 \sqrt{2}}=\frac{1}{\sqrt{2}} \end{array}$

The derivative of $f(\tan x)$ w.r.t. $g(\sec x)$ at $x=\frac{\pi}{4}$, where $f^{\prime}(1)=2$ and…

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