Question

If $y={\left(\tan x\right)}^{{\left(\tan x\right)}^{\tan x}}$, then find $\frac{dy}{dx}$ at $x=\frac{\pi }{4}$.

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

The correct option is A $0$

Taking $\log$ on both sides, we get
$\log y={\left(\tan x\right)}^{\tan x}\log \tan x$
Again taking $\log$ on both the sides
$\log \log y=\left[\tan x\log \tan x\right]+\log \log \tan x$
Differentiating w.r.t. $x$, we get
$\frac{1}{y\log y}\frac{dy}{dx}=\log \tan x\frac{d}{dx}\left(\tan x\right)+\tan x\frac{d}{dx}\left(\log \tan x\right)+\frac{d}{dx}\left(\log \log \tan x\right)$
$\frac{dy}{dx}=\log \tan x.{\sec }^{2}x+\tan x.\frac{{\sec }^{2}x}{\tan x}+\frac{{\sec }^{2}x}{\tan x.\log \tan x}$
$\frac{dy}{dx}=y\log y.{\sec }^{2}x\left[\log \tan x+1+\frac{1}{\tan x.\log \tan x}\right]$
At $x=\frac{\pi }{4}$, $y=1$ and $\log y=0$
So, putting this values in the equation of $\frac{dy}{dx}$, we get
$\therefore {\left(\frac{dy}{dx}\right)}_{x=\frac{\pi }{4}}=0$