Question

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

• A. $\frac{1}{\sqrt{2}}$
• B. $\sqrt{2}$
• C. $1$
• D. none of these

Answer 1

The correct option is A $\frac{1}{\sqrt{2}}$

$y=f\left(\tan x\right)$ and $z=g\left(\sec x\right)$

$\frac{dy}{dx}=f^{\prime }\left(\tan x\right).{\sec }^{2}x$ and $\frac{dz}{dx}=g^{\prime }\left(\sec x\right).\sec x\tan x$

$\therefore \frac{dy}{dz}=\frac{f^{\prime }\left(\tan x\right).{\sec }^{2}x}{g^{\prime }\left(\sec x\right).\sec x\tan x}$

${\begin{array}{c}\frac{dy}{dz}\end{array}|}_{x=\frac{\pi }{4}}=\frac{f^{\prime }\left(1\right).(\sqrt{2}{)}^2}{g^{\prime }\left(\sqrt{2}\right).\left(\sqrt{2}\right)}=\frac{1}{\sqrt{2}}$