Question

If $y=(\tan x{)}^{(\tan x{)}^{(\tan x{)}^{\cdots \infty }}}$, then find the value of $\frac{dy}{dx}$ at $x=\frac{\pi }{4}$

Answer 1

$y={\tan x}^{{\tan x}^{{\tan x}^{................\infty }}}$ .......(i)

From (i), we have

$y={\tan x}^y$
Taking $\log$ on both sides

$\log y=y\log \tan x$

Differentiating on both sides, we get

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

Put $x=\frac{\pi }{4}$ and $y=1$

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

$\Rightarrow \frac{dy}{dx}=\frac{2}{1}+\log 1\frac{dy}{dx}$

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

$\Rightarrow \frac{dy}{dx}=2$

So, correct answer is $2$.