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. $\sqrt{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $1$
• D. $\frac{1}{2}$

Answer 1

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

$\frac{df\left(\tan x\right)}{d\left(g\right(\sec x\left)\right)}=\frac{f^{\prime }\left(\tan x\right)se{c}^{2}x}{g^{\prime }\left(\sec x\right)\cdot \sec x\tan x}$

$\frac{df\left(\tan x\right)}{d\left(g\right(\sec x\left)\right)}at(x=\frac{\pi }{4})=\frac{f^{\prime }\left(1\right)\cdot \sqrt{2}}{g^{\prime }\left(\sqrt{2}\right)\cdot }=\frac{2\sqrt{2}}{4}=\frac{1}{\sqrt{2}}$