Question

If $y=\left[\right(\tan x{)}^{\tan x}{]}^{\tan x}$, then $\frac{dy}{dx}$ at $x=\pi /4$ is

• A. $2$
• B. $1/2$
• C. $1$
• D. $0$

Answer 1

Answer: (A)

$2$

Consider the following differential eq.

$y={\left({\left(\tan x\right)}^{\tan x}\right)}^{\tan x}\&x=\frac{\pi }{4}$

Taking $log$ both side, we get.

$\log y={\tan }^{2}x\log \left(\tan x\right)$

Differentiate both side w.r.t $x$, we get.

$\frac{1}{y}\frac{dy}{dx}=2\tan x{\sec }^{2}x\log \left(\tan x\right)+{\tan }^{2}x\frac{1}{\tan x}{\sec }^{2}x$

$\frac{1}{y}\frac{dy}{dx}={\left({\left(\tan x\right)}^{\tan x}\right)}^{\tan x}(2\tan \frac{\pi }{4}{\sec }^2\frac{\pi }{4}\log \left(\tan \frac{\pi }{4}\right)+\tan \frac{\pi }{4}{\sec }^2\frac{\pi }{4})$

$\frac{dy}{dx}=0+2$

$=2$

Hence, this is the required answer.